In calculus, a limit describes the behavior of a function as its input approaches a specific value, focusing on what the function gets infinitely close to, rather than its value at the exact point itself. Limits are fundamental because they are used to define continuity, derivatives, and integrals, which are the core concepts of calculus. 💡Key aspects of limits: • Approaching a value: Limits look at what value a function "approaches" as the input gets closer and closer to a specific number, from both the left and the right sides. • Not the function's value: The limit doesn't care about the function's actual value at the specific input point; it only cares about the behavior of the function near that point. • Fundamental to calculus: Limits are the building blocks for understanding continuity, the instantaneous rate of change (derivatives), and the area under a curve (integrals). 💡How limits work: • One-Sided Approach: To find a limit, you examine the function's behavior as the input approaches the target value from the left (values smaller than the target) and from the right (values larger than the target). • Existence: A limit only exists if the function approaches the same value from both the left and the right. • Notation: Limits are represented with the notation: lim f(x): means "the limit of the function f(x)". x → a: means "as x approaches a". For example, lim (x → a) f(x) = L is read as "the limit as x approaches a of f(x) equals L" 💡Why they matter: • Defining Continuity: A function is continuous at a point if its limit at that point equals the function's actual value there. • Defining Derivatives: The concept of a limit is used to define the derivative, which measures the instantaneous rate of change of a function. • Defining Integrals: Limits are also used in the definition of definite integrals, which represent the area under a curve. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1IfdCaeJTszq4Is48tDhUUxlMd-w9Eltx/view?usp=drive_link • Answers: https://drive.google.com/file/d/11PKq7Z-aEJQOqR_xbqFdAEQdQH4JjKnu/view?usp=drive_link 💡Chapters: 00:00 Why limits? Change at an instant. 01:08 Definition of a limit 01:57 Evaluate limits using graphs 03:27 Evaluate limits numerically using tables 04:39 Algebraic properties of limits 06:11 Evaluating limits using algebra 08:09 Examples on selecting methods for evaluating limits 13:24 Squeeze theorem 17:01 Examples on connecting representations of limits 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://www.youtube.com/channel/UCJAvCW22EeE_2s2ZlJne7uQ?sub_confirmation=1 _______________________ ⏩Playlist Link: https://www.youtube.com/playlist?list=PLm_WLG6GdV3vHdHSwImjsL4qzNf_9Qwlh _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://rumble.com/user/DrOfEng

In calculus, a function is continuous if its graph can be drawn without lifting your pen, meaning there are no breaks or jumps. More formally, a function f(x) is continuous at a point a if three conditions are met: f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a). 💡Intuitive Understanding • No Gaps or Jumps: You can draw the graph of a continuous function without your pencil leaving the paper. • Small Input, Small Output: A small change in the input (x) results in a small change in the output (f(x)). 💡The Formal Definition (at a Point a) For a function f(x) to be continuous at a point a, all three of the following conditions must be true: • f(a) is defined: The function must have a specific, defined value at the point a. • The limit of f(x) as x approaches a exists: The function must approach the same value as x gets closer and closer to a from both the left and right sides. • The limit equals the function value: The value the function approaches (the limit) must be the same as the actual value of the function at a, i.e., lim x→a f(x) = f(a). 💡Discontinuities If any of these three conditions are not met, the function is discontinuous at that point. Common types of discontinuities include: • Removable: A "hole" in the graph that could be "filled". • Jump: The graph jumps from one value to another, like the function f(x) = {-1 for x less than or equal to 0 and 1 for x greater than 0 at x=0}. • Infinite: A vertical asymptote where the function's value goes to positive or negative infinity. 💡Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1IfdCaeJTszq4Is48tDhUUxlMd-w9Eltx/view?usp=drive_link - Answers: https://drive.google.com/file/d/11PKq7Z-aEJQOqR_xbqFdAEQdQH4JjKnu/view?usp=drive_link 💡Chapters: 00:00 Types of discontinuities 01:58 Discontinuities, examples 04:26 Continuity at a point, with examples 08:16 Continuity over an interval, with example 10:55 Removing discontinuities, with example 13:42 Infinite limits and vertical asymptotes 14:41 Infinite limits and horizontal asymptotes 15:34 Relative magnitudes of functions 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains the Intermediate Value Theorem, including formula, statement, meaning, conditions, application to analysis for determining if a root exists, loose visual proof, walkthrough example exercises and practice problems and solutions (PDF). 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1IfdCaeJTszq4Is48tDhUUxlMd-w9Eltx/view?usp=drive_link - Answers: https://drive.google.com/file/d/11PKq7Z-aEJQOqR_xbqFdAEQdQH4JjKnu/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Intermediate Value Theorem definition 01:02 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://www.youtube.com/channel/UCJAvCW22EeE_2s2ZlJne7uQ?sub_confirmation=1 What part of Limits and Continuity gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://www.youtube.com/playlist?list=PLm_WLG6GdV3vHdHSwImjsL4qzNf_9Qwlh _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains what the meaning of the derivative is in maths (typically class 11 and 12 calculus), including introduction, formula, definition by first principle, how to represent and estimate it graphically, numerically, analytically, verbally and using technology. Worked example practice problems are provided. You will be able to use the definition to prove the derivatives of elementary functions like sin, cos, tan, ln, e to the power x, ex, x and trigonometric functions. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1PHEth8jm90Yy-42wKfBXG6_-1oi2nUDL/view?usp=drive_link - Answers: https://drive.google.com/file/d/1wlHR_SPWlSvvi0RjTmYAaxscP6WPTr-l/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Rate of change at a point 01:35 Definition of the derivative, with example 04:09 Representing the derivative 06:21 Derivatives at a point 07:40 Estimate the derivative using technology 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://www.youtube.com/channel/UCJAvCW22EeE_2s2ZlJne7uQ?sub_confirmation=1 What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://www.youtube.com/playlist?list=PLm_WLG6GdV3vHdHSwImjsL4qzNf_9Qwlh _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how differentiability implies continuity (3 conditions) of a function (e.g. composite or complex), including a basic introduction and visual proof using graphs and the limit formula, how to know if a function is differentiable at a point or when a function is not differentiable by first principles, and worked example practice questions. This topic is typically covered in class 11 or 12 maths. Understanding this topic is crucial to the chain, product and quotient rules, evaluating the derivatives of functions like sin, e^x, log and applying other differentiation rules. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-IfxXvhpMp43qxNG-21ZtIPqO4WMuvOk/view?usp=drive_link - Answers: https://drive.google.com/file/d/1nchFi76DuCzBIx0TWesDVjlUz0qvNG7o/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Differentiability and continuity 02:03 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to apply derivative rules like the power rule in a few minutes, which you can add to your Calculus 1 and 2 cheat sheet and apply to differentiate logarithmic (ln), exponential (e), trigonometric (trig) and other functions. A basic review and proof of the power law formula are given, which is applied in examples (PDF) where the exponents have fractions. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-IfxXvhpMp43qxNG-21ZtIPqO4WMuvOk/view?usp=drive_link - Answers: https://drive.google.com/file/d/1nchFi76DuCzBIx0TWesDVjlUz0qvNG7o/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Power rule proof, binomial theorem 01:33 Derivative rules 03:17 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://www.youtube.com/channel/UCJAvCW22EeE_2s2ZlJne7uQ?sub_confirmation=1 What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://rumble.com/user/drofeng

This video explains how to derive the derivatives of common elementary functions using first principles, which you can add to your Calculus 1 and 2 cheat sheet or formula sheet, including basic logarithmic (ln), exponential (e) and trigonometric (sin and cos). A worksheet (PDF) of worked example practice problems is provided. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-IfxXvhpMp43qxNG-21ZtIPqO4WMuvOk/view?usp=drive_link - Answers: https://drive.google.com/file/d/1nchFi76DuCzBIx0TWesDVjlUz0qvNG7o/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Derivative of cos(x) 01:24 Derivative of sin(x) 02:50 Derivative of e^x 03:37 Derivative of ln(x) 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to derive the product rule using first principles, which is commonly taught in class 11, 12 or A level maths. It is used extensively in physics and you can add it to your formula sheet or cheat sheet. The other important rules to be covered later are the quotient rule and chain rule. A worksheet (PDF) of worked example practice questions (problems) with solutions is provided, which also shows how the product rule can be applied to functions with three terms. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-IfxXvhpMp43qxNG-21ZtIPqO4WMuvOk/view?usp=drive_link - Answers: https://drive.google.com/file/d/1nchFi76DuCzBIx0TWesDVjlUz0qvNG7o/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Product rule derivation 01:42 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to apply the quotient rule formula for differentiating quotients of two functions (like uv, u and v). which is covered in class 11, 12 and A level maths. It can be applied to evaluate the derivatives of quotients of radicals, polynomials, logarithms and so on. The proof is deferred until the chain rule is covered. Worked example practice questions with solutions are provided. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-IfxXvhpMp43qxNG-21ZtIPqO4WMuvOk/view?usp=drive_link - Answers: https://drive.google.com/file/d/1nchFi76DuCzBIx0TWesDVjlUz0qvNG7o/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Quotient rule, mnemonic 00:38 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to evaluate the derivatives of trigonometric or circular functions using the product and quotient rules. You can memorise these derivatives and add them to your formula sheet. Trig identities are also applied to simplify the derivatives. Worked examples (PDF) with solutions are provided. The derivatives of inverse trig functions are deferred as they require the chain rule. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-IfxXvhpMp43qxNG-21ZtIPqO4WMuvOk/view?usp=drive_link - Answers: https://drive.google.com/file/d/1nchFi76DuCzBIx0TWesDVjlUz0qvNG7o/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Derivative of tan(x) 01:05 Derivative of cot(x) 02:17 Derivative of sec(x) 03:34 Derivative of cosec(x) 04:47 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Definition and Fundamental Properties) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to derive (step-by-step basic proof) and apply the chain rule to evaluate the derivatives of composite functions (u v), which is typically covered in class 11 and 12 maths. You can also add this to your formula sheet, along with the product rule and quotient rule. We will use it later to evaluate the derivatives of trig functions, and it can be extended to partial derivatives and integration (u-substitution). Worked example practice questions and solutions are provided to supplement your learning from platforms like Khan Academy. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1Kz_Al0UcyEqyLsrD1taFXvqQps-bHDR4/view?usp=drive_link - Answers: https://drive.google.com/file/d/11133Xv_5h2foZW0DxqseyDPMIR09aeB3/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Chain rule derivation, composite functions 01:39 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Composite, Implicit and Inverse Functions) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains and reviews the difference between implicit vs explicit functions and what the implicit method of differentiation is, when to use it to find dy/dx using the chain rule, and how to differentiate xy, ln, e, sin and so on. You can remember the method or add it to your formula sheet. Worked example questions or problems (PDF) with solutions are provided for practice. Later on, the applications will be covered, such as finding the equation of the tangent line. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1Kz_Al0UcyEqyLsrD1taFXvqQps-bHDR4/view?usp=drive_link - Answers: https://drive.google.com/file/d/11133Xv_5h2foZW0DxqseyDPMIR09aeB3/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Implicit differentiation, implicit vs explicit functions 01:15 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Composite, Implicit and Inverse Functions) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains the differentiation rule or equation and notation for inverse functions (dx dy), where the theorem and formula are commonly taught in class 12 maths. This topic is essential for obtaining inverse trig derivatives. A worksheet (PDF) of example problems is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1Kz_Al0UcyEqyLsrD1taFXvqQps-bHDR4/view?usp=drive_link - Answers: https://drive.google.com/file/d/11133Xv_5h2foZW0DxqseyDPMIR09aeB3/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Derivatives of inverse functions 01:45 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Composite, Implicit and Inverse Functions) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains the differentiation of inverse trig functions using trig identities, which is commonly taught in class 12 maths. The equations for the derivatives of the inverse of 6 trig functions are summarised and tips are provided on how to remember them. A worksheet (PDF) of example problems is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1Kz_Al0UcyEqyLsrD1taFXvqQps-bHDR4/view?usp=drive_link - Answers: https://drive.google.com/file/d/11133Xv_5h2foZW0DxqseyDPMIR09aeB3/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Derivatives of inverse trig functions, derivation 01:30 Worked example 02:46 Summary table 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/@drofeng What part of Differentiation (Composite, Implicit and Inverse Functions) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to obtain higher-order derivatives like the second derivative, including definition, formula, implicit differentiation using the chain rule and trigonometric functions. This topic is covered in class 12 maths, engineering maths, bsc 1st year and so on. A worksheet of questions or problems (PDF) and solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1Kz_Al0UcyEqyLsrD1taFXvqQps-bHDR4/view?usp=drive_link - Answers: https://drive.google.com/file/d/11133Xv_5h2foZW0DxqseyDPMIR09aeB3/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Higher-Order Derivatives 01:10 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Composite, Implicit and Inverse Functions) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to use some of the 7 rules of logarithms to differentiate logarithmic and exponential functions (e.g. dy/dx for log 4x), as well as the product and chain rules, which are covered in class 12 maths. The formula for simplifying the quotient rule is also covered, and the primary purpose and when to use logarithmic differentiation is made clear. A worksheet (PDF) of questions and answers is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1Kz_Al0UcyEqyLsrD1taFXvqQps-bHDR4/view?usp=drive_link - Answers: https://drive.google.com/file/d/11133Xv_5h2foZW0DxqseyDPMIR09aeB3/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Logarithmic differentiation 01:08 Differentiating exponential functions 02:07 Quotient rule 03:12 Worked examples 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Differentiation (Composite, Implicit and Inverse Functions) gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to interpret and understand the meaning of the derivative in contextual applications using simple words (what does it mean in context?). This applies to both first and second derivatives. The derivative is essentially the rate of change of one quantity with respect to another. A worksheet (PDF) of questions and answers is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1PHEth8jm90Yy-42wKfBXG6_-1oi2nUDL/view?usp=drive_link - Answers: https://drive.google.com/file/d/1wlHR_SPWlSvvi0RjTmYAaxscP6WPTr-l/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Derivative in context 01:15 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Contextual Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how an object's position, velocity and acceleration change over time in one direction or along a single axis (linear motion) using derivatives (rate of change or the slope of the tangent to a graphed curve). This topic is covered in class 11 maths (e.g. AP, NEET, IB) physics and engineering mechanics. Other than connecting position, displacement, velocity and acceleration, which are vector quantities with a magnitude or length and direction, scalar quantities like speed and distance are also covered. You will understand the difference between instantaneous and average quantities using the limit definition formula of the derivative, and when an object is speeding up or slowing down. A worksheet (PDF) of worked example practice problems and answers is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1PHEth8jm90Yy-42wKfBXG6_-1oi2nUDL/view?usp=drive_link - Answers: https://drive.google.com/file/d/1wlHR_SPWlSvvi0RjTmYAaxscP6WPTr-l/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Displacement vs distance 02:32 Velocity and speed 04:21 Acceleration 05:59 Connecting position, velocity and acceleration 07:24 Worked examples 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Contextual Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to solve related rates of change worded problems using the chain rule of differentiation, which is covered in class 12 maths (AP, IB, NEET). Related rates are hard because each problem has a different approach but this tutorial will help you learn to calculate them easily. The problems covered in the video and worksheet involve a cone, cylinder, ladder, area and volume. The worksheet (PDF) includes practice questions and solutions for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1PHEth8jm90Yy-42wKfBXG6_-1oi2nUDL/view?usp=drive_link - Answers: https://drive.google.com/file/d/1wlHR_SPWlSvvi0RjTmYAaxscP6WPTr-l/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Introduction to related rates 01:47 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Contextual Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how the local linearisation formula of a differentiable function simplifies calculations by estimating change, which is typically covered in class 12 maths (AP, IB, NEET). The definition, meaning and importance of the method is closely related to zooming in on the tangent line for estimating the value of the function near the given point. The tutorial also demonstrates how the differential method is used for error approximation. You will be able to understand if the error is overestimated or underestimated. A worksheet of problems with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1PHEth8jm90Yy-42wKfBXG6_-1oi2nUDL/view?usp=drive_link - Answers: https://drive.google.com/file/d/1wlHR_SPWlSvvi0RjTmYAaxscP6WPTr-l/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Local linearity 01:16 Error approximation 02:27 Worked examples 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Contextual Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains L'Hôpital's rule in simple terms, which is a method used to evaluate the limits of rational functions that may result in an indeterminate form (0/0 or infinity/infinity), including formula, proof, pronunciation, why it works and when you cannot apply it. L'Hospital's rule is covered class 11 and 12 maths, and engineering mathematics. The basic steps to apply L'Hopital's rule are to check for indeterminacy, find the derivatives, new quotient, evaluate the new limit, and repeat as necessary. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1PHEth8jm90Yy-42wKfBXG6_-1oi2nUDL/view?usp=drive_link - Answers: https://drive.google.com/file/d/1wlHR_SPWlSvvi0RjTmYAaxscP6WPTr-l/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 L'Hopital's rule, proof for indeterminate form 0/0 02:21 L'Hospital's rule, proof for indeterminate form infinity/infinity 06:03 Worked examples 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Contextual Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains what is the mean value theorem MVT for differentiation and how it is used in calculus. The MVT formula is covered in class 12 and engineering mathematics and helps with understanding derivatives. Continuity and differentiability are the that need to be satisfied for application of the MVT to first order and higher order derivatives. A worksheet (PDF) of questions with solutions is provided for practice, which includes applications to instantaneous and average velocity and second order derivatives. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 MVT definition and visual proof 01:01 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://www.youtube.com/channel/UCJAvCW22EeE_2s2ZlJne7uQ?sub_confirmation=1 What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://rumble.com/user/drofeng

This video explains the extreme value theorem for finding global (absolute) extrema and critical points (local or relative extrema) of functions on a closed interval, which are covered in basic calculus (calc 1, engineering mathematics). You will understand the meaning and why the topic is important through the conditions for extreme values in the definition and example graphs. The first and second derivative test will be covered later on, from which you will be able to distinguish critical points from inflection points. A worksheet (PDF) of problems with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Extreme value theorem 01:11 Critical points 02:09 Global and local extrema 03:17 Worked examples 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains the steps and gives the definition of the first derivative test for local maxima and minima, local extreme values of functions or relative extrema. A positive or negative first derivative tells us if a function is increasing/decreasing and identifies critical points. 1st, 2nd and 3rd derivatives of functions like trigonometric ones are calculated in a similar way and a table is typically used for common derivatives. The second derivative test will be covered later on. A worksheet (PDF) of problems with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Determine if a function is increasing/decreasing using the first derivative 01:11 Critical points 01:24 First derivative test 02:38 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains what is the candidates test for absolute extrema, which is covered in class 12 calculus (ap calc ab, bc). It is method for finding the absolute maximum and minimum values (absolute extrema) of a continuous function on a closed interval. You will understand the definition, steps and when to use it vs the first derivative. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Determine if a function is increasing/decreasing using the first derivative 01:11 Critical points 01:24 First derivative test 02:38 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains the second derivative test, which determines if a critical point of a function is a local minimum or maximum (concavity), or inflection point (= 0). The topic is covered in class 12 maths. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Concavity of functions 01:17 Second derivative test 02:17 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to sketch the graphs of original functions using relationships to their derivatives and limits. You will understand the method for matching and connecting the graphs using equations and tables. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Graphs of functions 01:28 Worked examples on sketching graphs 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to connect functions to their first and second derivative graphs by analysing the slope of the tangent to the curve of the original function. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Connecting functions and their first and second derivatives 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how optimisation is used to find the minimum or maximum values of a function (e.g. quadratic), which is used in real-life applications like minimising cost or maximising area. It is typically covered in grade 12 maths (calc 1, ap). The key steps are to reduce the problem to a function of one variable and find a critical point using differentiation. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Intro to optimisation 01:56 Worked example 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This video explains how to analyse the behaviour of implicit functions using implicit differentiation to find the derivatives (dy/dx and y''(x)). Like explicit functions, the derivatives reveal certain properties like increasing/decreasing, horizontal and vertical tangents, critical points and extrema. A worksheet (PDF) of questions with solutions is provided for practice. 📌 Worksheets are provided in PDF format to further improve your understanding: - Questions Worksheet: https://drive.google.com/file/d/1-6NeYDqLKVXwGn2cLGddt-RgPwE6BRkS/view?usp=drive_link - Answers: https://drive.google.com/file/d/1ZcFgkcsWJvJAyThTUTCfGbEP0LqGLGtx/view?usp=drive_link 📌 Chapters on What You’ll Learn: 00:00 Behaviour of implicit relations 01:54 Worked examples 📌 Perfect for calculus students and exam preparation, this tutorial uses plenty of examples and easy-to-follow explanations. 👉 Make sure to check out our full Calculus 1 and 2 course playlist, which covers differentiation, integration, sequences and series and their applications! – Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔴Subscribe: https://rumble.com/user/drofeng What part of Analytical Applications of Differentiation gave you the most difficulty? Comment below!👇 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

"Accumulation of change" is a fundamental calculus concept referring to the total change of a quantity over a specific interval, calculated by integrating its rate of change. For example, if you have the velocity of a car (the rate of change of position), the accumulated change in position (the total distance traveled or displacement) is found by integrating the velocity function over a given time period. This concept is represented graphically as the area under the curve of the rate function within the specified interval. 💡Key Aspects • Rate of Change: Accumulation of change works with the rate at which a quantity changes. • Integration: The mathematical operation used to find the accumulation of change is definite integration. • Net Change: The result is the total, or net, change in the quantity over the interval. • Graphical Representation: The definite integral of a rate function is equivalent to the area between the curve of that function and the x-axis over a specific interval. 💡Real-World Examples • Distance Traveled: Integrating a car's velocity over time gives the total distance it has traveled. • Water in a Tank: Integrating the rate of water flow into a tank gives the total volume of water accumulated over a period. • Revenue Earned: Integrating the rate of daily earnings gives the total revenue over a week. 💡Distinction from Total Value • The accumulation of change tells you how much a quantity has changed, but not its total value. • To find the total value, you need a boundary value—the starting or ending value of the quantity at a specific point. You then add this boundary value to the accumulated change. For instance, if you know the initial amount of water in a tank and add the total water accumulated (from integration) to it, you get the final volume of water. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Accumulation of change from the rate function 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

A Riemann sum approximates the area under a curve by dividing the region into a series of rectangles and summing their areas. The width of each rectangle is a subinterval, and its height is determined by evaluating the function at a specific point within that subinterval, such as the left endpoint, right endpoint, or midpoint. By increasing the number of rectangles, the Riemann sum becomes a more accurate approximation of the exact area under the curve, which is found by taking the limit as the number of rectangles approaches infinity. 💡How Riemann Sums Work • Divide the Interval: The interval over which you want to find the area is divided into smaller, equally-sized subintervals. • Construct Rectangles: For each subinterval, a rectangle is formed with the x-axis as its base. • Determine Rectangle Height: The height of each rectangle is found by evaluating the function at a chosen point within its subinterval. Common choices include: ◦ Left Endpoint Rule: The height is determined by the function's value at the left-most point of the subinterval. ◦ Right Endpoint Rule: The height is determined by the function's value at the right-most point of the subinterval. ◦ Midpoint Rule: The height is determined by the function's value at the midpoint of the subinterval. • Calculate Rectangle Area: The area of each individual rectangle is calculated by multiplying its width (the subinterval's width) by its height. • Sum the Areas: The Riemann sum is the total of the areas of all these rectangles. 💡The Relationship to Integration • A Riemann sum is an approximation of the definite integral. As the number of rectangles increases, the approximation becomes more precise. When the number of rectangles approaches infinity, the Riemann sum converges to the exact area under the curve, which is the value of the definite integral. 💡Purpose of Riemann Sums • Approximation: Riemann sums provide a way to estimate the area under a curve when an exact antiderivative might be difficult or impossible to find. • Foundation of Integration: They serve as a fundamental concept in calculus, illustrating how a continuous function's area can be broken down and added up. • Building Intuition: By visualizing the process of subdivision, approximation, and summation, Riemann sums help in understanding the underlying principles of integral calculus. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Intro to Riemann sums 01:26 Left Riemann sum 02:42 Right Riemann sum 03:33 Midpoint Riemann sum 04:47 Trapezoidal sum 06:20 Worked examples 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

A definite integral calculates the net signed area between a function's curve and the x-axis over a specific interval, defined by a lower limit (a) and an upper limit (b) of integration. The definite integral of a function f(x) from a to b, written as ∫ᵇₐ f(x) dx, is found by calculating the antiderivative F(x) of f(x) and then evaluating F(b) - F(a). The result is a single number representing the exact area, which can be positive, negative, or zero depending on whether the area is above, below, or crosses the x-axis. 💡Key Components and Notation • Integral Symbol (∫): Indicates the operation of integration. • Integrand (f(x)): The function being integrated. • Limits of Integration (a and b): The lower (a) and upper (b) bounds of the interval over which the area is calculated. • Differential (dx): Indicates the variable of integration. 💡How to Evaluate a Definite Integral • Find the Antiderivative: Determine the indefinite integral (antiderivative) F(x) of the function f(x). • Substitute Limits: Substitute the upper limit (b) and the lower limit (a) into the antiderivative F(x). • Subtract: Subtract the result from the lower limit substitution from the result of the upper limit substitution: F(b) - F(a). 💡Example To find the definite integral of 2x from 1 to 2: • Antiderivative: The antiderivative of 2x is x². • Evaluate at Limits: ◦ Upper limit: (2)² = 4 ◦ Lower limit: (1)² = 1 • Subtract: 4 - 1 = 3. • Therefore, ∫²₁ 2x dx = 3. 💡When and Why to Use Definite Integrals Definite integrals are used to: • Calculate Exact Areas: Find the precise area under a curve over a given interval. • Measure Net Signed Area: Determine the total area above the x-axis minus the total area below the x-axis within the interval. • Model Accumulation: Measure the total buildup of a quantity over a period when its rate of change is known, such as the total distance traveled by a car given its velocity function over time. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Definite integral, definition 01:38 Integral to Riemann sum 03:33 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The Fundamental Theorem of Calculus, Part 1, states that if a function f is continuous on an interval [a, b], then the function G(x) defined by the integral of f(t) from a to x (G(x) = ∫_a^x f(t)dt) is continuous on [a, b], differentiable on (a, b), and its derivative, G'(x), is equal to f(x). Essentially, taking the derivative of this integral "undoes" the integration, resulting in the original function f. 💡Key Aspects • Inverse Operations: This part of the theorem demonstrates that differentiation and integration are inverse operations. • Antiderivative: The integral G(x) defined in this way is an antiderivative of f(x). • Conditions: The theorem requires the function f to be continuous on the specified interval. • The Integral as a Function: The definite integral of f(t) from a constant a to a variable x results in a new function G(x), not just a single numerical value. 💡In simpler terms: If you define a function G(x) by finding the area under the curve of another function f(t) from a fixed point a up to a variable point x, then the rate at which that area changes (the derivative of G(x)) is simply the height of the curve f(x) at that point x. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Fundamental theorem of calculus, part 1, visual proof 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Accumulation functions represent the running total, or the cumulative area, under the graph of another function, often over a fixed interval starting at a constant point 'a' and extending to a variable point 'x'. For a given function f(t), the accumulation function F(x) = ∫[from a to x] f(t) dt measures this total accumulated quantity, such as distance traveled or total rainfall. The First Fundamental Theorem of Calculus shows that the accumulation function F(x) is an antiderivative of f(x), meaning its rate of change, F'(x), is equal to the original function f(x). 💡Key Characteristics • Definition: An accumulation function F(x) is defined as a definite integral where the upper limit is a variable, such as F(x) = ∫[from a to x] f(t) dt. • Meaning: It provides a cumulative measure of the quantity represented by f(t) over the interval from 'a' to 'x'. • Relationship to the Original Function: According to the First Fundamental Theorem of Calculus, the derivative of an accumulation function F(x) is the original function f(x). • Behavior: ◦ F(x) increases when f(x) is positive. ◦ F(x) decreases when f(x) is negative. ◦ F(x) has a maximum or minimum value when f(x) = 0, representing points where the area accumulation changes direction. • Applications: Accumulation functions are used to model quantities that change over time or intervals, such as: ◦ Total distance traveled by a car. ◦ Total amount of water in a tank after a certain time. ◦ Total rainfall over a given period. 💡Example Consider the function f(t) = 2t representing the velocity of an object at time t. The accumulation function for distance, D(x), would be: D(x) = ∫[from 0 to x] 2t dt This function D(x) gives the total distance the object has traveled from time 0 to time x, and D'(x) = 2x, which is the original velocity function. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Fundamental theorem of calculus, part 1, visual proof 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Definite integral properties in calculus simplify evaluating integrals and calculating the net signed area under a curve. Key properties include: the integral is zero when limits are the same (\int_a^a f(x)dx = 0), reversing limits changes the sign (\int_a^b f(x)dx = -\int_b^a f(x)dx), an integral of a sum is the sum of integrals (\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx), a constant factor can be pulled out (\int_a^b kf(x)dx = k\int_a^b f(x)dx), and an integral can be split into two intervals (\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx). 💡Basic Properties • Zero Integral: If the upper and lower limits of integration are the same, the definite integral is zero. ◦ Formula: \int_a^a f(x)dx = 0 ◦ Explanation: The interval has no width, and thus no area is enclosed. • Reverse Limits: If you swap the upper and lower limits of integration, the sign of the definite integral reverses. ◦ Formula: \int_a^b f(x)dx = -\int_b^a f(x)dx ◦ Explanation: This signifies a change in the accumulated area's direction. 💡Properties for Calculation • Sum/Difference of Functions: The integral of a sum or difference of functions is the sum or difference of their individual integrals. ◦ Formula: \int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx ◦ Explanation: This property allows you to break down complex integrals into simpler ones. • Constant Multiple: A constant factor multiplied by a function within an integral can be factored out. ◦ Formula: \int_a^b kf(x)dx = k\int_a^b f(x)dx (where 'k' is a constant) ◦ Explanation: Constants can be pulled out of the integral to simplify the integration process. • Additive Interval Property: An integral over an interval can be split into the sum of two integrals over sub-intervals. ◦ Formula: \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx ◦ Explanation: This property is valid for any value of 'c', not just when it lies between 'a' and 'b'. 💡Other Useful Properties • Variable Substitution: A definite integral's value is independent of the variable of integration. ◦ Formula: \int_a^b f(x)dx = \int_a^b f(t)dt ◦ Explanation: You can replace the variable of integration with another variable, like 't', without changing the outcome. • Integration of Even and Odd Functions: Properties exist for integrating even or odd functions over symmetric intervals around the origin, which can simplify calculations. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Definite integrals, properties 01:35 Integrals over discontinuities 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The Second Fundamental Theorem of Calculus states that for a continuous function f on the interval [a, b], the definite integral from a to b is equal to the difference between the value of any antiderivative F at b and the value of F at a. Its formula is: ∫baf(x)dx = F(b) − F(a), where F'(x) = f(x). This theorem provides a powerful method for evaluating definite integrals by finding an antiderivative and evaluating it at the integral's limits. 💡Here's a breakdown of the theorem: • The Setup: You start with a function f(x) that is continuous on the interval [a, b]. • The Antiderivative: You find an antiderivative of f(x), which is any function F(x) such that its derivative, F'(x), is equal to f(x). • The Calculation: The theorem says that the definite integral, which represents the area under the curve of f(x) from a to b, can be calculated by taking the antiderivative F(x), evaluating it at the upper limit b (F(b)), and then subtracting the value of the antiderivative evaluated at the lower limit a (F(a)). 💡Why is it important? • Simplifies Calculation: It provides a direct method to calculate definite integrals, replacing the more complex process of breaking the area into infinite tiny rectangles (numerical integration). • Connects Integration and Differentiation: It establishes a fundamental link between the process of integration and differentiation by showing that integrating a function's derivative gives you back the original function (up to a constant difference). • Applications: It is widely used in various fields to find total changes by integrating a rate of change. For example, if f(x) is a velocity function, the integral of f(x) from a to b gives the total displacement over that time interval. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Fundamental theorem of calculus, part 2 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

An indefinite integral, also known as an antiderivative, is a function whose derivative is the original function. It represents a family of functions that differ by a constant, which is included as a "+ C" in the final result, denoted by the integral symbol (∫) followed by the function and a differential (like dx). For example, the indefinite integral of f(x) is F(x) + C, where F'(x) = f(x). 💡Key Characteristics • Antiderivative: It is the inverse operation of differentiation. • Constant of Integration (+ C): This constant is crucial because the derivative of any constant is zero. • No Fixed Boundaries: Unlike definite integrals, indefinite integrals do not have upper and lower limits of integration. • Symbolic Representation: The notation ∫f(x)dx means "find the antiderivative of f(x)". The "dx" indicates the variable of integration and is the differential form of the function. 💡Example To find the indefinite integral of f(x) = x²: • Recall the power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. • Apply the rule to x²: ∫x² dx = x²⁺¹/(2+1) + C = x³/3 + C. • The result, x³/3 + C, represents all functions whose derivative is x², such as x³/3, (x³/3) + 5, or (x³/3) - 10. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Indefinite integrals, power rule 01:28 Elementary antiderivatives 02:40 Antiderivatives of tan, cot, sec, cosec 04:15 Antiderivatives of inverse trig functions 05:34 Antiderivatives of log and exponential functions 07:00 Worked examples 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Integration by substitution is a calculus technique to solve integrals of composite functions, acting as the reverse of the chain rule for differentiation. The method involves substituting a part of the integrand with a new variable (often u) to simplify the integral. You identify a function and its derivative within the integrand, set the function as u, then find du (the derivative of u with respect to x multiplied by dx). This transforms the integral into a simpler form, ∫f(u)du, which can be integrated using basic rules. Finally, you substitute the original function back for u to get the antiderivative in terms of the original variable. 💡Steps for Integration by Substitution • Identify the substitution: Look for an inner function, g(x), whose derivative, g'(x), is also a factor in the integrand. • Choose u: Let u = g(x). • Find du: Differentiate u with respect to x to find du/dx, then rearrange to du = g'(x) dx. • Rewrite the integral: Substitute u and du into the original integral. • Integrate with respect to u: Solve the new, simpler integral using standard integration rules. • Substitute back: Replace u with the original function g(x) to get the final answer. 💡Example Let's integrate ∫ 2x cos(x²) dx: • Identify: The function cos(x²) has an inner function x², and its derivative is 2x, which is also in the integral. • Choose u: Let u = x². • Find du: du/dx = 2x, so du = 2x dx. • Rewrite: ∫ 2x cos(x²) dx becomes ∫ cos(u) du. • Integrate: ∫ cos(u) du = sin(u) + C. • Substitute back: Replace u with x² to get sin(x²) + C. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Integration by substitution 01:12 u-substitution, definite integrals 02:08 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Integration using long division is a technique to integrate improper rational functions, where the degree of the numerator is greater than or equal to the degree of the denominator. The process involves performing polynomial long division to rewrite the integrand as a polynomial plus a proper rational function (where the numerator's degree is less than the denominator's). The polynomial part is integrated directly using basic rules, while the proper rational function part is then integrated using other methods, such as partial fraction decomposition or completing the square. 💡When to Use Long Division for Integration • Improper Rational Functions: Use this method when you're faced with an integral containing a rational expression (a fraction of polynomials) where the numerator's degree is greater than or equal to the denominator's degree. 💡Steps for Integration Using Long Division • Perform Long Division: Divide the numerator by the denominator to find the quotient and remainder. • Rewrite the Integrand: Express the original rational function as the quotient plus the remainder divided by the divisor. ∘ Original integrand = Quotient + (Remainder / Divisor) • Integrate Term by Term: The integral can now be broken down into the integral of the polynomial (quotient) and the integral of the proper rational function (remainder/divisor). ∘ ∫ (Numerator/Denominator) dx = ∫ (Quotient) dx + ∫ (Remainder/Divisor) dx • Integrate the Polynomial: The integral of the polynomial quotient is found using the power rule for integration. • Integrate the Proper Rational Function: The remaining integral, which is now a proper rational function, can be integrated using techniques like: • Partial Fraction Decomposition: If the denominator can be factored into simpler linear or irreducible quadratic factors. • Completing the Square: If the denominator is an irreducible quadratic, completing the square can transform it into a form suitable for arctangent integrals or logarithmic forms. • U-Substitution: U-substitution might be used to simplify the integral of the proper rational piece. 💡Example To integrate ∫ (x³ + 3x² + 8x + 19) / (x² + 5) dx, you would first perform long division. • Long Division: Divide x³ + 3x² + 8x + 19 by x² + 5, yielding a quotient of (x + 3) and a remainder of (3x + 4). • Rewrite: The integral becomes ∫ (x + 3 + (3x + 4)/(x² + 5)) dx. • Integrate: ∘∫ (x + 3) dx = x²/2 + 3x ∘∫ (3x / (x² + 5)) dx + ∫ (4 / (x² + 5)) dx ∘These remaining integrals are solved using u-substitution and arctangent integration, respectively, resulting in (3/2)ln(x² + 5) and (4/√5)arctan(x/√5). 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Integration by polynomial long division 01:02 Worked Example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Integration by completing the square is a method for evaluating integrals with quadratic expressions in the denominator, particularly when the quadratic is irreducible (cannot be factored into real linear terms). The process involves rewriting the quadratic \(ax^{2}+bx+c\) into the form \(a(x+k_{1})^{2}+k_{2}\), which is then manipulated using u-substitution into a standard integral form, typically involving the inverse tangent (\(\arctan \)) or inverse sine (\(\arcsin \)) functions. 💡Steps for Completing the Square • Ensure x² coefficient is one: If the coefficient of \(x^{2}\) (let's call it \(a\)) is not 1, factor it out from the entire quadratic expression. ∘ For example, \(2x^{2}-4x+11\) becomes \(2(x^{2}-2x+\frac{11}{2})\). • Complete the square on the remaining quadratic: Take the coefficient of the \(x\) term (let's call it \(b^{\prime }\)), divide it by 2, and then square the result. Add and subtract this value inside the parenthesis. ∘ For example, in \(x^{2}-2x+\frac{11}{2}\), the \(x\) coefficient is -2. ∘ Half of -2 is -1, and squaring it gives 1. ∘ The expression becomes \(2(x^{2}-2x+1+\frac{11}{2}-1)\). • Factor the perfect square trinomial: The first three terms inside the parenthesis now form a perfect square trinomial, \((x+b^{\prime }/2)^{2}\). ∘ \(2((x-1)^{2}+\frac{9}{2})\). • Simplify the constant term: Combine the remaining constant terms. ∘ \(2((x-1)^{2}+\frac{9}{2})=2(x-1)^{2}+9\). 💡Applying to Integrals • Rewrite the denominator: Substitute the completed square form into the integral. ∘ For example, \(\int \frac{1}{2x^{2}-4x+11}dx\) becomes \(\int \frac{1}{2(x-1)^{2}+9}dx\). • Use substitution: Let \(u\) be the term in the parenthesis. In the example, \(u=x-1\), so \(du=dx\). ∘ The integral becomes \(\int \frac{1}{2u^{2}+9}du\). • Manipulate to match standard forms: Factor out constants and rearrange the expression to match a known integral, often involving the form \(\frac{1}{a^{2}+u^{2}}\). ∘ \(\int \frac{1}{2u^{2}+9}du=\frac{1}{2}\int \frac{1}{u^{2}+(3/\sqrt{2})^{2}}du\). • Integrate: Apply the appropriate integral formula. ∘ The integral \(\int \frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan (\frac{x}{a})+C\). ∘ The final result can be converted back to the original variable. 💡When to Use This Method • Irreducible quadratics: This method is essential for integrals involving quadratic denominators that do not factor into real linear terms. • Non-standard forms: It is useful for integrands that resemble natural logarithms or inverse trigonometric functions but are "off" due to the quadratic expression. • Standard forms: Completing the square transforms a general quadratic into a form that can be solved using known integration techniques, such as those found in integral tables. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Completing the square 00:40 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Integration by parts is a calculus technique to integrate the product of two functions, derived from the product rule for differentiation. It transforms a complex integral into simpler ones, using the formula: ∫u dv = uv - ∫v du. To apply it, you select a function to be 'u' (which will be differentiated) and the rest as 'dv' (which will be integrated), often guided by the LIATE mnemonic to choose 'u' such that its derivative simplifies, and 'dv' such that its integral is easily found. 💡The Formula The integration by parts formula is: ∫u dv = uv - ∫v du 💡When to Use It This method is particularly useful for integrating products of functions, such as: • A logarithmic function multiplied by an algebraic function. • A trigonometric function multiplied by an algebraic function. • An algebraic function multiplied by an exponential function. 💡Steps to Apply the Formula • Identify u and dv: From the product of functions to be integrated, choose a function 'u' and the remaining part as 'dv'. • Find du and v: Differentiate 'u' to find 'du' and integrate 'dv' to find 'v'. • Apply the formula: Substitute 'u', 'v', 'du', and 'dv' into the integration by parts formula. • Simplify: Evaluate the new, hopefully simpler, integral. 💡How to Choose u and dv (The LIATE Rule) The acronym LIATE can help decide which function to choose for 'u': • L: ogarithmic functions (e.g., ln(x)) • I: nverse trigonometric functions (e.g., arcsin(x)) • A: lgebraic functions (e.g., x, x²) • T: rigonometric functions (e.g., sin(x), cos(x)) • E: xponential functions (e.g., e^x) The function that appears earliest in this list should generally be chosen as 'u'. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Integration by parts 01:05 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Integration using partial fractions is a method to solve integrals of complex rational functions by breaking them down into simpler fractions whose integrals are easier to find, typically using logarithms. The process involves decomposing the integrand into partial fractions, solving for the unknown constants in the numerator of these fractions, and then integrating each simplified fraction. 💡Steps for Integration by Partial Fractions • Ensure Proper Rational Function: The degree of the numerator must be less than the degree of the denominator. If it's not, perform polynomial long division first to obtain a proper fraction. • Factor the Denominator: Factor the denominator of the rational function into its simplest components, such as distinct linear factors, repeated linear factors, irreducible quadratic factors, and repeated irreducible quadratic factors. • Set up the Partial Fraction Decomposition: Write the original rational function as a sum of partial fractions, with appropriate numerators over each factor. ∘ For a distinct linear factor (ax+b), the term is A/(ax+b). ∘ For a repeated linear factor (ax+b)ⁿ, the terms are A₁/(ax+b) + A₂/(ax+b)² + ... + Aₙ/(ax+b)ⁿ. ∘ For an irreducible quadratic factor (ax²+bx+c), the term is (Ax+B)/(ax²+bx+c). • Solve for the Constants: Equate the original numerator with the sum of the numerators of the partial fractions. Solve this equation for the unknown constants (A, B, etc.) by either substituting convenient values for x or by comparing coefficients of like powers of x. • Integrate the Simpler Fractions: Substitute the found partial fractions back into the integral and integrate each term separately. ∘ Integrals of the form ∫(A/(ax+b)) dx typically result in A/a * ln|ax+b| + C. ∘ Add the constant of integration, C, for indefinite integrals. 💡Example • To integrate (6x+13)/(x²+5x+6), you would first factor the denominator as (x+2)(x+3). Then, you would set up the decomposition: (6x+13)/((x+2)(x+3)) = A/(x+2) + B/(x+3). • Solving for A and B gives A = 7 and B = -1. • So the integral becomes: ∫(7/(x+2) - 1/(x+3)) dx • This can be integrated as: 7 ln|x+2| - ln|x+3| + C 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Integration using partial fractions, distinct factors 01:24 Repeated factors 03:42 Irreducible quadratic 05:21 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Improper integrals extend definite integrals to situations with an infinite integration interval or a function with a discontinuity within the interval. They are evaluated by replacing the problematic limit (infinity or a discontinuity) with a variable, say 't', and then calculating the definite integral from that variable to the other limit. The key is to take the limit as the variable 't' approaches the infinite or problematic value. If the limit results in a finite number, the integral converges; if the limit does not exist or is infinite, the integral diverges. 💡Two Types of Improper Integrals • Type 1: Infinite Limits of Integration ∘Description: This type involves integrating over an interval that extends to infinity, such as from a finite number to positive or negative infinity, or between two infinite limits. ∘Example: The integral of 1/x² from 1 to infinity: ∫₁^∞ (1/x²) dx. ∘Evaluation: ⬩Replace the infinite limit with a variable (e.g., 'b'): lim (b→∞) ∫₁^b (1/x²) dx. ⬩Evaluate the definite integral: ∫₁^b (1/x²) dx = [-1/x]₁^b = -1/b - (-1/1) = 1 - 1/b. ⬩Take the limit as 'b' approaches infinity: lim (b→∞) (1 - 1/b) = 1. ⬩Conclusion: Since the limit is a finite number (1), the integral converges. • Type 2: Integrands with a Discontinuity ∘Description: This type of integral occurs when the function being integrated has a vertical asymptote or some other form of discontinuity within the interval of integration. ∘Example: The integral of 1/x from 0 to 1: ∫₀¹ (1/x) dx, where x=0 is a point of discontinuity. ∘Evaluation: ⬩The approach from the left for the discontinuity. ⬩Replace the problematic limit (0) with a variable (e.g., 'a') and take the limit: lim (a→0⁺) ∫ₐ¹ (1/x) dx. ⬩Evaluate the definite integral: ∫ₐ¹ (1/x) dx = [ln|x|]ₐ¹ = ln|1| - ln|a| = 0 - ln|a| = -ln|a|. ⬩Take the limit as 'a' approaches 0 from the right: lim (a→0⁺) (-ln|a|) = ∞. ⬩Conclusion: Since the limit is infinity, the integral diverges. 💡Convergence vs. Divergence • Convergent: An improper integral is convergent if the limit of the corresponding definite integral exists and is a finite number. • Divergent: An improper integral is divergent if the limit does not exist or is infinite. By using limits, improper integrals allow us to determine if an unbounded area under a curve or an area with a discontinuity can be represented by a finite value. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Improper integrals 01:31 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

To select the right integration technique, start by simplifying the integrand and trying a simple u-substitution. If that doesn't work, identify the integral's type: a product of functions suggests integration by parts or tabular integration, while rational functions may require partial fractions. Other advanced methods include trigonometric substitution for expressions involving sums or differences of squares, and trigonometric integrals for simplifying trig functions. If the initial method fails, consider trying a combination of techniques. 💡General Steps for Choosing an Integration Technique • Simplify the Integrand: Before anything else, try to simplify the function you are integrating using algebraic manipulation or trigonometric identities. • Try a Simple Substitution (u-Substitution): Look for a function and its derivative (or something that can be made into its derivative with a constant multiplier) within the integrand. • Identify the Form of the Integral: ∘Product of Functions: If you have a product, consider Integration by Parts. Tabular integration is a fast version of this if one factor's derivative eventually becomes zero. ∘Rational Functions: If you have a rational function (a polynomial divided by another polynomial), partial fraction decomposition is often effective. ∘Sums/Differences of Squares: For expressions like \(a^{2}\pm x^{2}\) or \(x^{2}\pm a^{2}\), consider a trigonometric substitution. ∘Trigonometric Functions: Look for patterns in powers of sine and cosine or use identities to simplify. • Apply the Appropriate Technique: If a simple substitution doesn't work, move to the more advanced techniques based on the identified form. • Try Combinations: Complex integrals may require a sequence of different techniques. For example, you might use u-substitution to simplify the integrand, then integration by parts. 💡When to Use Specific Techniques • U-Substitution: Use for integrals where the derivative of a part of the integrand is also present. • Integration by Parts: Use when you have a product of two functions, especially when u-substitution fails. • Partial Fractions: Use for integrating rational functions where the denominator can be factored. • Trigonometric Substitution: Use for integrals containing expressions like \(\sqrt{a^{2}-x^{2}}\), \(\sqrt{a^{2}+x^{2}}\), or \(\sqrt{x^{2}-a^{2}}\). • Trigonometric Integrals: Use for simplifying integrals involving powers of sine and cosine by using trigonometric identities. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link • Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link 💡Chapters: 00:00 Selecting integration techniques 02:02 Worked examples 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

A differential equation is an equation that involves an unknown function and its derivatives, describing how a quantity changes over time or space. These equations are fundamental to modeling real-world phenomena in fields like physics, biology, and engineering, where change is a central concept. The solution to a differential equation is not a single number but a function or set of functions that satisfy the equation's conditions. 💡What is a differential equation? • An equation containing an unknown function (e.g., y = f(x)) and one or more of its derivatives. • For example, dy/dx represents the rate of change of y with respect to x. 💡How are they used? • Modeling real-world changes: They are used to describe processes like population growth, heat transfer, radioactive decay, and the motion of objects. • A language of science: They provide a powerful mathematical framework for expressing the laws of physics and other scientific principles. 💡Key Concepts • Function and Derivatives: The equation establishes a relationship between the function itself and its rate of change (its derivative). • Solution is a function: Unlike algebraic equations, the solution to a differential equation is a function or a family of functions, not a specific value. • Initial Conditions: To find a unique, "particular" solution, you need additional information called an initial condition (e.g., the starting position or velocity of an object). 💡Examples • Newton's Law of Cooling: Describes how the temperature of an object changes over time as it cools down. • Population Dynamics: Models how the size of a population changes based on factors like birth and death rates. • Simple Harmonic Motion: Describes the oscillation of a mass attached to a spring, involving the second derivative of its position. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Intro to differential equations 01:00 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

To verify a solution to a differential equation, take the derivatives of the proposed solution as needed to match the order of the differential equation. Then, substitute the original function and its derivatives into the original differential equation. If the left side of the equation equals the right side, then the proposed function is a valid solution. 💡Steps to Verify a Solution • Identify the proposed solution and the differential equation. • Calculate the necessary derivatives: of the proposed solution. For example, if the differential equation involves a first derivative, calculate the first derivative of the function; if it involves a second derivative, calculate both the first and second derivatives. • Substitute the function and its derivatives: into the original differential equation. • Simplify both sides of the equation . • Check for consistency . If the left-hand side (LHS) of the equation is equal to the right-hand side (RHS), the function is a solution. If LHS does not equal RHS, it is not a solution. 💡Example Consider the differential equation dy/dx = 3x^2 and the proposed solution y = x^3. • Identify: ⚬ Differential equation: dy/dx = 3x^2 ⚬ Proposed solution: y = x^3 • Calculate derivatives: ⚬ dy/dx = 3x^2 (using the power rule). • Substitute: ⚬ Replace dy/dx in the differential equation with the calculated derivative: 3x^2 = 3x^2. • Simplify: ⚬ The equation is already in its simplest form. • Check for consistency: ⚬ 3x^2 = 3x^2 is true for all values of x, so y = x^3 is a verified solution. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Verifying solutions to ODEs 01:18 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

A slope field is a visual tool for a first-order differential equation of the form dy/dx = f(x, y), where short tangent lines (or segments) are drawn at many points (x, y) on a graph. Each segment's slope is determined by the value of f(x, y) at that specific point, providing a visual representation of the gradient of all possible solution curves to the differential equation without explicitly finding the solutions themselves. 💡How to Construct a Slope Field • Identify the differential equation: You'll have an equation like dy/dx = f(x, y). • Choose points: Select a grid of (x, y) coordinates on the xy-plane. • Calculate the slope at each point: Substitute the x and y values into f(x, y) to find the slope (dy/dx) at that point. • Draw the segments: At each chosen point, draw a short line segment whose slope matches the calculated value. • Observe the field: The collection of all these segments creates the slope field. 💡What a Slope Field Shows • Direction of solutions: The segments show the direction of the tangent lines to any solution curve at that point. • Qualitative behavior: It gives a general idea of how the solution curves behave without providing the explicit formula for the solutions. • Solution curves: By "following the flow" of the segments, you can sketch a particular solution curve for a given initial condition. 💡Applications Slope fields are useful for studying differential equations that are difficult or impossible to solve analytically, allowing for the visualization of their solutions and behaviors. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Sketching slope fields, with example 02:09 Reasoning and sketching solution curves, with example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Euler's method is a simple numerical technique for approximating solutions to first-order ordinary differential equations (ODEs) with an initial value. It works by iteratively moving along the tangent line at each step, using the derivative to find the slope, and advancing the solution by a small step size ('h') to estimate the next point on the solution curve. The basic formula is y₁ = y₀ + h * f(x₀, y₀), where y₀ is the initial value, x₀ is the initial x-value, h is the step size, and f(x₀, y₀) is the slope at the starting point derived from the differential equation. 💡How it Works • Problem Setup: You start with a differential equation in the form y' = f(x, y) and an initial condition (x₀, y₀). • Choose a Step Size (h): This is the small increment you'll use to move from one point to the next along the x-axis. • Calculate the Slope: At the current point (xₙ, yₙ), find the slope using the differential equation: m = f(xₙ, yₙ). • Estimate the Next Point: Move from the current point by taking a small step of size 'h' along the tangent line. The new y-value is calculated as: yₙ₊₁ = yₙ + h * f(xₙ, yₙ). • Repeat: Use the new point (xₙ₊₁, yₙ₊₁) as your starting point for the next step and repeat the process to find the next approximation. 💡Key Formula The core of Euler's method is the iterative formula: • **yₙ₊₁ = yₙ + h • f(xₙ, yₙ)** 💡Example Application Imagine you want to find the approximate value of a solution to y' = 2x and y(1) = 3, using a step size of h = 0.1. • Initial Condition: (x₀, y₀) = (1, 3). • Step 1: ⚬ Calculate f(x₀, y₀) = f(1, 3) = 2 * 1 = 2. ⚬ Calculate y₁ = y₀ + h * f(x₀, y₀) = 3 + 0.1 * 2 = 3.2. • Step 2: ⚬ The new point is (x₁, y₁) = (1 + 0.1, 3.2) = (1.1, 3.2). ⚬ Calculate f(x₁, y₁) = f(1.1, 3.2) = 2 * 1.1 = 2.2. ⚬ Calculate y₂ = y₁ + h * f(x₁, y₁) = 3.2 + 0.1 * 2.2 = 3.42. You continue this process to find subsequent points. 💡Accuracy and Limitations • Step Size (h): The accuracy of Euler's method depends on the step size. Smaller step sizes generally lead to better approximations but require more computational steps. • First-Order Method: It is a first-order method, meaning it assumes the slope remains constant over the small interval, which introduces some error. • Applications: Despite its limitations, Euler's method is fundamental in computational science and forms the basis for more complex numerical methods used to solve and simulate differential equations in fields like engineering and physics. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Euler's method 01:23 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The separation of variables method is a mathematical technique for solving differential equations by algebraically rearranging them so that all terms involving one variable are on one side of the equation and all terms involving the other variable are on the opposite side. This separation allows each side of the equation to be integrated independently, leading to an equation without derivatives, which represents the solution to the original differential equation. The method is applicable to differential equations that can be expressed in a product form, such as dy/dx = g(x)h(y). 💡How it Works • Rewrite the Equation: The first step is to rewrite the differential equation so that all terms containing the dependent variable (e.g., y and dy) are on one side, and all terms containing the independent variable (e.g., x and dx) are on the other. For example, if you have dy/dx = ky, you would rewrite it as dy/y = kdx. • Integrate Both Sides: Once the variables are separated, integrate both sides of the equation. Remember to include a constant of integration on one side. • Solve for the Variable: Simplify the integrated equation and, if possible, solve for the dependent variable to find the explicit solution. 💡Example To solve the differential equation dy/dx = ky: • Separate variables: Multiply by dx and divide by y to get dy/y = kdx. • Integrate: Integrate both sides: ∫(1/y) dy = ∫k dx. • Solve: This yields ln|y| + C₁ = kx + C₂, which simplifies to ln|y| = kx + C (where C = C₂ - C₁). Taking the exponential of both sides gives y = e^(kx+C) = Ae^(kx) (where A = e^C), which is the general solution. 💡Key Considerations • Applicability: The method is only applicable if the differential equation can be expressed in a specific product form, like dy/dx = g(x)h(y). • Implicit Solutions: It's not always possible to find an explicit solution; you may have to leave the solution in an implicit form. • Singular Solutions: The method might miss certain singular solutions that don't fit the general form of the solution obtained. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Separation of variables, general solution, with example 02:40 Particular solution, with examples 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://www.youtube.com/channel/UCJAvCW22EeE_2s2ZlJne7uQ?sub_confirmation=1 _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Exponential models describe quantities that change at a rate proportional to their current value, which is captured by the differential equation dy/dt = ky. The solution to this equation is y(t) = y(0)e^(kt), where 'y(t)' is the quantity at time 't', 'y(0)' is the initial quantity, 'k' is the constant of proportionality, and 'e' is the base of the natural logarithm. If 'k' is positive, the model represents exponential growth; if 'k' is negative, it represents exponential decay. 💡How it works • The Differential Equation: The core idea is that the rate of change of a quantity (dy/dt) is directly proportional to the quantity itself (y), with 'k' as the constant of proportionality. • The Solution (Exponential Model): By solving this separable differential equation, we arrive at the formula for exponential growth or decay: ⚬ y(t) = y(0)e^(kt) ⚬ y(t): The amount of the quantity at time 't'. ⚬ y(0): The initial amount of the quantity at time 't' = 0. ⚬ k: The growth or decay constant. ⚬ e: The base of the natural logarithm, approximately 2.71828. 💡Applications This relationship is widely applied in various fields: • Population Growth: The rate at which a population grows is often proportional to its current size. • Radioactive Decay: The rate at which a radioactive substance decays is proportional to the amount of the substance remaining. • Finance: Continuously compounded interest follows an exponential model, where the growth of money is proportional to the amount present. • Medicine: The elimination of a drug from the bloodstream can often be modeled using exponential decay. 💡Interpreting the constant 'k' • Positive 'k': Indicates exponential growth, where the quantity increases at an accelerating rate. • Negative 'k': Indicates exponential decay, where the quantity decreases at a rate proportional to its current value. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Exponential models, growth and decay 01:40 Worked examples 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The logistic growth differential equation is represented as dP/dt = rP(1 - P/K), where P is the population, t is time, r is the growth rate, and K is the carrying capacity, which represents the maximum population an environment can sustain. This equation models how a population's growth slows down as it approaches the carrying capacity, providing a more realistic model than simple exponential growth. 💡Components of the Equation • dP/dt: The rate of change of the population with respect to time. • P: The population size at any given time t. • r: The intrinsic growth rate, which determines how quickly the population would grow without environmental limitations. • K: The carrying capacity, the maximum population size the environment can sustainably support. 💡How the Equation Works • When P is small: The term (1 - P/K) is close to 1, so dP/dt ≈ rP, resembling exponential growth. • As P approaches K: The term (1 - P/K) becomes smaller, reducing the growth rate dP/dt. • When P = K: The term (1 - P/K) becomes 0, and dP/dt = 0, indicating that the population has stabilized at the carrying capacity. This model contrasts with exponential growth, where the growth rate continuously increases with the population size. The logistic equation accounts for limited resources, ensuring that the population growth eventually levels off. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link • Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link 💡Chapters: 00:00 Logistic growth model 01:15 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The Mean Value Theorem for Integrals states that for a continuous function \(f(x)\) on a closed interval \([a,b]\), there is at least one point \(c\) within the interval where the function's value \(f(c)\) equals the function's average value over the interval, given by \(\frac{1}{b-a}\int _{a}^{b}f(x)\,dx\). Mathematically, this is expressed as \(f(c)=\frac{1}{b-a}\int _{a}^{b}f(x)\,dx\), or equivalently, \(\int _{a}^{b}f(x)\,dx=f(c)(b-a)\). This means a rectangle with width \((b-a)\) and height \(f(c)\) will have the same area as the definite integral of \(f(x)\) over the interval [a, b]. 💡Key Aspects of the Theorem • Continuity is Required: The function \(f(x)\) must be continuous on the closed interval \([a,b]\). • Existence of 'c': The theorem guarantees that at least one such point \(c\) exists within the interval, but it does not provide a method to find \(c\). • Average Value: The value \(f(c)\) represents the average height of the function over the interval. • Geometric Interpretation: Geometrically, the theorem states that for a continuous function, there exists a rectangle with height \(f(c)\) and width \((b-a)\) that has the same area as the region under the curve of \(f(x)\) from \(a\) to \(b\). 💡Mathematical Statement • If \(f(x)\) is continuous on the closed interval \([a,b]\), then there exists a number \(c\in [a,b]\) such that: \(f(c)=\frac{1}{b-a}\int _{a}^{b}f(x)\,dx\) • Or, this can be rearranged to: \(\int _{a}^{b}f(x)\,dx=f(c)(b-a)\) 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Mean value theorem for integrals 01:00 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Position, velocity, and acceleration are linked by calculus: velocity is the derivative of position (rate of change), and acceleration is the derivative of velocity (rate of change of velocity), or the second derivative of position. To find the related function, you can differentiate to go forward (position to velocity to acceleration) or integrate to go backward (acceleration to velocity to position). 💡Derivatives (Going Forward) • Position (\(s(t)\)) to Velocity (\(v(t)\)): Velocity is the instantaneous rate of change of position with respect to time. To find the velocity function, you differentiate the position function. \(v(t)=\frac{d}{dt}s(t)\) • Velocity (\(v(t)\)) to Acceleration (\(a(t)\)): Acceleration is the instantaneous rate of change of velocity with respect to time. To find the acceleration function, you differentiate the velocity function. \(a(t)=\frac{d}{dt}v(t)\) • Position (\(s(t)\)) to Acceleration (\(a(t)\)): Acceleration is also the second derivative of the position function with respect to time. \(a(t)=\frac{d^{2}}{dt^{2}}s(t)\) 💡Integrals (Going Backward) • Acceleration (\(a(t)\)) to Velocity (\(v(t)\)): To find the velocity from acceleration, you take the antiderivative (integral) of the acceleration function and add a constant of integration. \(v(t)=\int a(t)\,dt\) • Velocity (\(v(t)\)) to Position (\(s(t)\)): To find the position from velocity, you take the antiderivative (integral) of the velocity function and add a constant of integration, which represents the initial position. \(s(t)=\int v(t)\,dt\) 💡In Summary • Position: Where an object is located. • Velocity: How fast the object is changing its position and in what direction. • Acceleration: How fast the object is changing its velocity. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Connecting position, velocity and acceleration 01:59 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

Interpreting definite integrals in context involves recognizing that the integral of a rate function over an interval provides the net change in the quantity the rate describes, which is equivalent to the area under the rate curve. For example, the definite integral of a velocity function gives the net displacement, and the integral of a production rate gives the total amount produced. 💡To Interpret a Definite Integral • Identify the Rate Function: Determine what the integrand (the function being integrated) represents. This is typically a rate of change, such as velocity, production rate, or revenue rate. • Identify the Quantity Accumulating: Understand what the rate function describes. For example, if the rate is in "gallons per minute," the accumulated quantity is "gallons". • Identify the Interval: The limits of integration (the numbers at the top and bottom of the integral symbol) define the time period or interval over which the net change is calculated. • Apply the Net Change Concept: The definite integral from 'a' to 'b' of a rate function, \(r(t)\), gives the net change in the quantity from time 'a' to time 'b'. This is found by the Fundamental Theorem of Calculus: \(\int _{a}^{b}r(t)dt=F(b)-F(a)\), where \(F(t)\) is the antiderivative (or amount function) of \(r(t)\). • Consider the Area Under the Curve: The definite integral also represents the area under the curve of the rate function over the specified interval. 💡Example Scenarios • Distance and Velocity: If \(v(t)\) is the velocity of an object, then \(\int _{a}^{b}v(t)dt\) represents the net change in the object's position (displacement) between time \(t=a\) and \(t=b\). • Total Production: If \(P(t)\) is the rate of production, then \(\int _{0}^{4}P(t)dt\) represents the net amount of product produced between \(t=0\) and \(t=4\) hours. • Revenue: If \(R(t)\) is the rate of revenue, then \(\int _{1}^{5}R(t)dt\) gives the net change in revenue from month 1 to month 5. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Integrals in applied contexts, accumulation functions 01:26 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The area between two curves, f(x) and g(x), is found by integrating the difference between the upper and lower functions over a given interval [a, b]: Area = ∫[a to b] (f(x) - g(x)) dx. This method works by integrating the area under the upper curve and subtracting the area under the lower curve, similar to finding the area of a square with a circular hole. For functions of y, the formula becomes Area = ∫[y1 to y2] (u(y) - v(y)) dy, where u(y) is the rightmost function and v(y) is the leftmost. 💡Steps to find the area between two curves: • Identify the functions: Determine the equations of the two curves, often represented as f(x) and g(x). • Find the limits of integration: Determine the points where the curves intersect to establish the interval [a, b]. If a specific interval is given, use that. • Determine the upper and lower functions: Identify which function has a greater y-value over the given interval. • Set up the integral: Write the definite integral of the upper function minus the lower function. ⚬ For integration with respect to x: Area = ∫[a to b] (f(x) - g(x)) dx. ⚬ For integration with respect to y: Area = ∫[y1 to y2] (u(y) - v(y)) dy. • Evaluate the integral: Calculate the definite integral to find the area. The resulting area will always be a positive value. 💡Example: To find the area between y = x and y = x² from x=0 to x=1: • Functions: f(x) = x, g(x) = x². • Limits: The interval is given as. • Upper/Lower: For 0 lt x lt 1, x gt x². So, f(x) is the upper function. • Integral: Area = ∫[0 to 1] (x - x²) dx. • Evaluate: [1/2 x² - 1/3 x³] from 0 to 1 = (1/2 - 1/3) - (0) = 1/6. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Area between curves, functions of x, with example 01:50 Area between curves, functions of y, with example 04:32 Area between curves with intersections, with example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

In calculus, volumes using cross-sections are found by integrating the area of a slice across a region, expressed as \(V=\int _{a}^{b}A(x)\,dx\) or \(V=\int _{c}^{d}A(y)\,dy\). To use this method, you must: 1) Determine the base of the solid and the shape of its cross-sections perpendicular to an axis (e.g., squares, rectangles, triangles), 2) Find the formula for the area, \(A\), of one cross-section in terms of the integration variable (e.g., \(A(x)\)), and 3) Integrate this area function over the appropriate bounds to sum the volumes of all the infinitesimal slices, yielding the total volume. 💡Steps for Calculating Volume with Cross-Sections • Sketch the Solid and Its Base: Start by sketching the region that forms the base of the solid and the shape of its cross-sections. • Identify the Cross-Sectional Area Function: ⚬ Choose the variable of integration (either \(x\) or \(y\)). ⚬ Determine the formula for the area, \(A\), of a single cross-section based on the shape of that slice (e.g., \(A=s^{2}\) for a square). ⚬ Express this area, \(A(x)\) or \(A(y)\), as a function of only that chosen integration variable. • Determine the Limits of Integration: Find the upper and lower bounds (\(a\) and \(b\), or \(c\) and \(d\)) for your integration based on the region of the base. • Set up and Evaluate the Integral: ⚬ The volume is then given by the definite integral: \(V=\int _{a}^{b}A(x)\,dx\) or \(V=\int _{c}^{d}A(y)\,dy\). ⚬ Evaluate the integral to find the total volume of the solid. 💡Example: Square Cross-Sections • If the base is a region bounded by curves, and the cross-sections perpendicular to the x-axis are squares, the area \(A(x)\) of a cross-section is the square of the side length, \(s(x)\), found from the base. • So, \(A(x)=[s(x)]^{2}\). • The volume is then \(V=\int _{a}^{b}[s(x)]^{2}\,dx\), where \(a\) and \(b\) are the limits along the x-axis. 💡Key Concept: • Infinite Summation: This method works by approximating the volume as a sum of infinitely many thin slices (each with volume Area × thickness) and then using the definite integral to find the exact sum. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Volumes with cross sections, squares and rectangles, with example 02:41 Triangle and semicircle cross sections, with examples 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The disk method calculates the volume of a solid of revolution by integrating the areas of infinitesimally thin circular disks, which are stacked to form the solid. The volume of a single disk is its cross-sectional area, \(\pi r^{2}\), multiplied by its thickness (either \(dx\) or \(dy\)). The total volume is found by integrating this expression over the limits of the solid's extent along the axis of rotation, yielding the formula \(V=\int _{a}^{b}\pi [r(x)]^{2}dx\) when rotating around the x-axis. 💡How the Disk Method Works • Visualize the Solid: Imagine a 2D region being rotated around an axis to create a 3D solid. Identify the Cross-Section: The cross-section of the solid, perpendicular to the axis of rotation, will be a circle (a disk). • Determine the Radius: The radius (\(r\)) of this disk is the distance from the axis of rotation to the outer boundary of the region being rotated. • Calculate the Area: The area of this circular disk is given by \(A=\pi r^{2}\). • Introduce Thickness: The thickness of the disk is an infinitesimal value, represented as \(dx\) if you are integrating along the x-axis or \(dy\) if you are integrating along the y-axis. • Formulate the Integral: The volume of the entire solid is found by summing (integrating) the volumes of an infinite number of these disks across the region's extent. 💡Formulas • Rotating around the x-axis: If a region is bounded by a function \(f(x)\) and the x-axis from \(x=a\) to \(x=b\), and rotated around the x-axis, the volume is: \(V=\int _{a}^{b}\pi [f(x)]^{2}dx\) • Here, \(r(x)=f(x)\) is the radius, and the thickness is \(dx\). • Rotating around the y-axis: If a region is bounded by a function \(g(y)\) and the y-axis from \(y=c\) to \(y=d\), and rotated around the y-axis, the volume is: \(V=\int _{c}^{d}\pi [g(y)]^{2}dy\) • Here, \(r(y)=g(y)\) is the radius, and the thickness is \(dy\). 💡Key Considerations • Axis of Rotation: The formula and the choice of \(dx\) or \(dy\) depend on whether the rotation is around the x-axis or the y-axis. • Radius Function: The radius must be expressed as a function of the variable of integration (x or y). • Integral Limits: The integral is evaluated between the specified limits of the region being rotated. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Volume with the disc method, revolve about x and y axes, with examples 04:11 Disc method, revolve about other axes, with example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The washer method calculates the volume of a solid of revolution with a hole by integrating the area of its "washers" (hollow discs). The volume of a single washer is found by subtracting the volume of the inner cylinder from the outer one, which is equivalent to \(\pi (R^{2}-r^{2})\Delta x\), where \(R\) is the outer radius, \(r\) is the inner radius, and \(\Delta x\) is the thickness. To find the total volume, you set up and solve a definite integral using this formula, where the limits of integration are determined by the intersection points of the functions bounding the region. 💡Steps for using the washer method • Sketch the region: Draw the region between the two functions and identify the axis of rotation. • Determine the representative rectangle: Draw a small, representative rectangle that is perpendicular to the axis of rotation. This rectangle will form a washer when rotated. ⚬ If the rectangle is vertical (perpendicular to the x-axis), the thickness is \(dx\), and you will integrate with respect to \(x\). ⚬ If the rectangle is horizontal (perpendicular to the y-axis), the thickness is \(dy\), and you will integrate with respect to \(y\). • Find the inner and outer radii (\(r\) and \(R\)): ⚬ Outer radius (\(R\)): The distance from the axis of rotation to the outer curve of the region. ⚬ Inner radius (\(r\)): The distance from the axis of rotation to the inner curve of the region. ⚬ This can be found by taking the "outer function" minus the "axis of rotation" (for vertical rectangles) or "right function" minus "axis of rotation" (for horizontal rectangles). Be sure both radii are expressed in terms of the same variable you are integrating with respect to. • Set up the integral: ⚬ The formula for the volume of a single washer is \(dV=\pi (R^{2}-r^{2})dx\) or \(dV=\pi (R^{2}-r^{2})dy\). ⚬ The total volume is the integral of this expression over the appropriate interval: \(V=\int _{a}^{b}\pi (R(x)^{2}-r(x)^{2})dx\) or \(V=\int _{c}^{d}\pi (R(y)^{2}-r(y)^{2})dy\). • Find the limits of integration: Determine the values of \(x\) or \(y\) where the inner and outer curves intersect to find the lower and upper bounds of your integral. • Evaluate the integral: Solve the definite integral to find the numerical volume of the solid. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Volume with the washer method, revolve around x and y axes, with example 03:55 Washer method, revolve around other axes, with example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

In calculus, arc length is the length of a curve between two points, calculated by integrating a formula that sums infinitesimal segments of the curve. For a smooth function \(y=f(x)\) over an interval \([a,b]\), the arc length \(L\) is found by integrating the square root of \((1+[f^{\prime }(x)]^{2})\) with respect to \(x\) from \(a\) to \(b\): \(L=\int _{a}^{b}\sqrt{1+[f^{\prime }(x)]^{2}}\,dx\). 💡Deriving the Arc Length Formula The formula is derived by approximating the curve with a series of tiny line segments and finding the limit as the number of segments approaches infinity. • Divide the curve: into \(n\) small segments. • Approximate the length: of each segment with the hypotenuse of a right triangle, where the legs are the change in \(x\) (\(\Delta x\)) and the change in \(y\) (\(\Delta y\)). • Use the Pythagorean theorem: The length of each segment is \(\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}\). • Rewrite: this in terms of the derivative \(\frac{dy}{dx}\): \(\sqrt{(\Delta x)^{2}(1+(\frac{\Delta y}{\Delta x})^{2})}=\Delta x\sqrt{1+(\frac{dy}{dx})^{2}}\). • Sum the lengths: of all these segments to approximate the total arc length: \(\sum \Delta x\sqrt{1+(\frac{dy}{dx})^{2}}\). • Take the limit: as \(\Delta x\) approaches 0, transforming the sum into a definite integral: \(\int _{a}^{b}\sqrt{1+(\frac{dy}{dx})^{2}}\,dx\). 💡Formulas for Different Cases • For \(y=f(x)\): If the curve is defined as a function of \(x\), the arc length \(L\) from \(x=a\) to \(x=b\) is: \(L=\int _{a}^{b}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx\quad \text{or}\quad L=\int _{a}^{b}\sqrt{1+[f^{\prime }(x)]^{2}}\,dx\). For \(x=g(y)\): If the curve is defined as a function of \(y\), the arc length \(L\) from \(y=c\) to \(y=d\) is: \(L=\int _{c}^{d}\sqrt{1+\left(\frac{dx}{dy}\right)^{2}}\,dy\quad \text{or}\quad L=\int _{c}^{d}\sqrt{1+[g^{\prime }(y)]^{2}}\,dy\). • For Parametric Curves: For a curve defined by parametric equations \(x=x(t)\) and \(y=y(t)\), the arc length \(L\) from \(t=a\) to \(t=b\) is: \(L=\int _{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt\). 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Arc length integral 02:04 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The "volume with shells" method, also known as the shell method, calculates the volume of a solid of revolution by integrating the volume of infinitesimally thin cylindrical shells. The formula is \(V=\int _{a}^{b}2\pi rh\,dx\) (for vertical rotation) or \(V=\int _{a}^{b}2\pi rh\,dy\) (for horizontal rotation), where \(r\) is the radius, \(h\) is the height of the shell, and \(dx\) or \(dy\) is the thickness. This method is often easier to use than the disk/washer method when the axis of rotation is perpendicular to the axis of the representative rectangle. 💡How to use the shell method • Visualize the solid: Imagine a region in the plane. When you revolve it around an axis, it forms a 3D solid. The shell method views this solid as being made of many thin cylindrical "shells" stacked together. • Determine the axis of rotation: ⚬ If rotating around a vertical axis (like the y-axis), you will integrate with respect to \(x\). The radius and height will be functions of \(x\). ⚬ If rotating around a horizontal axis (like the x-axis), you will integrate with respect to \(y\). The radius and height will be functions of \(y\). • Define the radius (\(r\)) and height (\(h\)): ⚬ Radius (\(r\)): This is the distance from the axis of rotation to the representative rectangle. ⚬ Height (\(h\)): This is the length of the representative rectangle, which is usually a function of the variable of integration. When revolving two functions, the height is the top function minus the bottom function (for vertical rotation) or the rightmost function minus the leftmost function (for horizontal rotation). • Set up the integral: Use the formula, which is the integral of the volume of a single shell (\(2\pi rh\,dx\) or \(2\pi rh\,dy\)) from the lower limit (\(a\)) to the upper limit (\(b\)) of your region. ⚬ Vertical axis rotation: \(V=\int _{a}^{b}2\pi xf(x)\,dx\), where \(x\) is the radius and \(f(x)\) is the height. ⚬ Horizontal axis rotation: \(V=\int _{a}^{b}2\pi yf(y)\,dy\). • Solve the integral: Calculate the definite integral to find the total volume. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1GQtYzPXeZpvnm7EdpOAPfEyr-VSltAse/view?usp=drive_link • Answers: https://drive.google.com/file/d/1urkbDfPaDPsymE8T58TZZJHgeUw-P-qx/view?usp=drive_link 💡Chapters: 00:00 Volume with the cylindrical shell method 01:26 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

The arc length (\(L\)) of a parametric curve given by \(x(t)\) and \(y(t)\) for \(t\) from \(a\) to \(b\) is found by integrating the square root of the sum of the squares of the derivatives of the x and y components with respect to \(t\), from \(a\) to \(b\), using the formula: \(L=\int _{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt\). 💡Here's how to calculate the arc length: • Find the derivatives: of the parametric equations with respect to \(t\). Let \(x^{\prime }(t)=\frac{dx}{dt}\) and \(y^{\prime }(t)=\frac{dy}{dt}\). • Square the derivatives: and add them together: \((x^{\prime }(t))^{2}+(y^{\prime }(t))^{2}\). • Take the square root: of this sum: \(\sqrt{(x^{\prime }(t))^{2}+(y^{\prime }(t))^{2}}\). • Integrate: the result with respect to \(t\) over the given interval, from \(a\) to \(b\). 💡Example: Consider the parametric curve \(x(t)=t^{2}\) and \(y(t)=t^{3}\) for \(t\) from \(0\) to \(1\). • Derivatives: \(\frac{dx}{dt}=2t\) and \(\frac{dy}{dt}=3t^{2}\). • Square and add: \((2t)^{2}+(3t^{2})^{2}=4t^{2}+9t^{4}\). • Integrate: \(L=\int _{0}^{1}\sqrt{4t^{2}+9t^{4}}\,dt=\int _{0}^{1}t\sqrt{4+9t^{2}}\,dt\). Using a u-substitution (\(u=4+9t^{2}\)), the integral evaluates to \(\frac{1}{27}(13\sqrt{13}-8)\). 💡Why use this formula? • It is derived from the Pythagorean theorem, where a small segment of the curve \(\Delta L\) is approximated by the hypotenuse of a small right triangle with sides \(\Delta x\) and \(\Delta y\). As \(\Delta t\rightarrow 0\), this becomes \(dL=\sqrt{(dx)^{2}+(dy)^{2}}\). The arc length integral sums up these infinitesimal lengths to find the total length of the curve. • It is especially useful for curves that cannot be easily expressed as a function of \(y\) in terms of \(x\), such as spirals. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1tolpJ9mIz-hrSc3BJyC0piUzPQ8cKD3b/view?usp=drive_link • Answers: https://drive.google.com/file/d/1tiM6iM6LTV0bkAhLvC8ij3bjvbloV5Jj/view?usp=drive_link 💡Chapters: 00:00 Parametric arc length integral 01:56 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe: https://rumble.com/user/drofeng _______________________ ⏩Playlist Link: https://rumble.com/playlists/Ptm8YeEDb_g _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://youtube.com/@drofeng

This calculus video covers a tutorial which provides an introduction to differential equations and modelling.

This calculus video covers a tutorial on verifying if a function is a solution to a differential equation.

This calculus video covers a tutorial on sketching slope fields for differential equations. 0:00 Intro to slope fields 1:01 Worked example on sketching slope fields

This calculus video covers a tutorial on sketching solution curves for differential equations using slope fields.

This video explains Euler's formula, including definition, step size, examples, and what it is used for. In this case, it is used to approximate solutions to first order ordinary differential equations.

This video explains the separation of variables method for solving differential equations, including example practice problems. You will understand how to separate the variables and know when to use this method.

This video explains the separation of variables method for solving differential equations, including example practice problems. You will understand how to separate the variables and know when to use this method. 0:00 Separation of variables, particular solution 2:04 Worked examples

This video explains exponential models with differential equations, including example applications like population growth, formula and what they predict. 0:00 Modelling exponential growth and decay using ODEs 1:41 Worked examples on exponential growth and decay

This video explains the logistic population growth mathematical model, including the differential equation, solution curve, function, graph, rate, and formula. 0:00 Intro to the logistic growth model 1:40 Finding a particular solution 2:53 Verifying the carrying capacity 3:54 Time at the maximum population growth rate

This video explains the mean value theorem (MVT) formula and definition for integrals using an informal geometric proof, what it is, how do you know if it applies and what it is used for.

This video explains the meaning of displacement and difference to distance, simple definition of speed and velocity, and how acceleration is calculated. 0:00 Connecting position, velocity and acceleration using graphs and integrals 2:17 Worked examples

The video explains what is and how to apply a definite integral in context with example problems. You will understand what does it mean in simple terms and real world situations that involve definite integrals, as well as the meaning of formulas of accumulation functions. 0:00 Definite integrals in applied contexts 1:35 Worked examples

How to find the area under the curve or between curves by integration. What the value (can be negative) or formula means or refers to is dependent on the application. Rectangles perpendicular to the x axis tell you that the integral is with respect to x. 0:00 Finding areas between curves using definite integrals 0:52 Worked examples

How to find the area under the curve or between curves by integration. What the value (can be negative) or formula means or refers to is dependent on the application. Rectangles perpendicular to the y axis tell you that the integral is with respect to y. 0:00 Finding areas between curves using definite integrals 1:25 Worked examples

Why the definite integral is the area under a curve or between two curves using examples. Understanding the meaning and properties of the formula or equation, if it gives the negative or net area, making appropriate substitutions for the limits of integration and how to solve area problems using the definite integral. 0:00 Finding areas between curves using definite integrals 1:35 Worked examples

How to find the volume of a prism with a constant cross-section or solid object with a cross section using integration. Basic geometry is used to derive the square cross-section formula. Worked examples and practice problems are covered for visualization. Square cross sections are covered while triangles (e.g. isosceles) and semicircles are deferred to another tutorial. 0:00 Finding the volume of solids with cross-sections using definite integrals 1:19 Worked examples

How to find the volume of a prism with a constant cross-section or solid object with a cross section using integration. Basic geometry is used to derive the triangle and semi-circle cross-section formula. Worked examples and practice problems are covered for visualization. Triangle (equilateral and isosceles) and semicircle cross sections are covered. 0:00 Volume of solids with triangle cross-sections 3:25 Volume of solids with semicircle cross-sections 4:37 Worked examples

Integration using the disc method to find the volume of solid of revolution (e.g. sphere and cone proof), and what it is. Examples show how to know when to use it (e.g. vs the washer or shell method), what is the R, integral formula about x axis and y axis, and derive the formula to calculate the volume of a sphere and cone. 0:00 Volume with disk method 1:53 Worked examples

Integration using the disc method to find the volume of solid of revolution, and what it is. Examples show how to know when to use it (e.g. vs the washer or shell method), what is the R, integral formula about other axes. 0:00 Volume with disk method 1:44 Worked example

This video explains the washer method vs the disk method, including what it is, when to use it, and how to find the radius. You will understand the difference to the disk method. The washer method around a line and shell method are deferred to another tutorial. 0:00 Volume with the washer method 1:29 Worked examples

This video explains the washer method to find the volume of a revolved solid around a vertical and horizontal line using integration. You will understand the difference to the disk method. 0:00 Volume with the washer method 1:42 Worked examples

This video explains the arc length integral formula in Cartesian coordinates, including examples on determining the arc length of a line and other curves. 0:00 Deriving the integral 2:06 Worked examples

This video explains how to use the shell method to calculate the volume of a solid of revolution, including integral formula, equation, practice example problems (x and y axis). You will understand the difference between the washer and shell method. 0:00 Volume with the shell method 1:41 Practice example problems

This video explains the meaning of parametric equations (eq), including what they are with example practice questions, the rule, how do you differentiate them and how to determine if the parametric form is differentiable. You will understand how they can be applied to determine the distance traveled and direction of motion. Converting an equation to parametric form is deferred to another tutorial. 0:00 Parametric equations, definition and differentiation 1:21 Worked examples

This video explains the meaning of parametric equations (eq), including what they are with example practice questions, the rule, how do you differentiate them (2nd derivative) and how to determine if the parametric form is twice differentiable. You will understand how they can be applied to determine the distance traveled and direction of motion. Converting an equation to parametric form is deferred to another tutorial. 0:00 Parametric equations, second derivative 1:13 Worked example

This video explains how to find the arc length of a parametric curve (e.g. y = 2x), including equations, formula and example. You will understand what it means in the context of distance traveled by a particle. 0:00 Parametric curve, arc length 1:59 Worked example

This video explains vector valued functions including definition, formula, problems and solutions, parameterization and graphing. You will understand what it is and how do you evaluate it. Motion in space is deferred to another tutorial. 0:00 Definition 1:43 Differentiation 3:47 Worked examples

This video explains how to integrate a vector valued function, including indefinite integral and practice problems. You will understand if you can integrate it and what it is. Derivatives and how do you turn a vector valued function into a scalar function are covered in other tutorials. 0:00 Integration of vector-valued function 1:32 Worked example

This video explains vector-valued functions in the context of solving motion problems, including position, velocity, speed and acceleration using differentiation, integration and arc length integrals, and worked example problems and solutions that involve projectile motion and uniform circular motion. 0:00 Vector-valued functions, solving motion problems 1:48 Worked examples

This video explains how to differentiate equations of polar curves, including equation, second derivative, and plotting polar curves like a cardioid, limacon and lemniscate. 0:00 Definition of polar coordinates 1:37 Polar curves 3:05 Differentiating polar equations 4:52 Worked examples, plotting polar curves 11:16 Worked examples, differentiating polar equations

This video explains how do you find the area of a region in a polar curve using integration, including formula, graph, derived integral, and worked example practice problem. 0:00 Polar curve area integral 1:52 Worked example

This video explains how to find the area of a region bounded by two polar curves (e.g. circle, limacon, lemniscate, cardioid) using integration, including formula, equations, and practice problems. 0:00 Area between two polar curves, integral 1:29 Worked examples

This video explains what is a conic section in polar form, including the derivation of the polar equation for an ellipse, hyperbola and parabola using the focus and directrix, practice problem, eccentricity, graphing, and focus at the pole or origin. 0:00 Revision on conic sections 2:06 Equation of a parabola 6:08 Equation of an ellipse 13:59 Equation of a hyperbola 20:15 Common polar equation 25:00 Worked example

This video explains the meaning of infinite sequences, including sum formula, definition, graphical representation, examples, epsilon-delta limit and notation. You will understand how to distinguish between infinite and finite sequences, and how to determine convergence. 0:00 Into to infinite sequences 1:02 Convergence 2:20 Example on epsilon-delta limit 4:04 Worked examples

This video explains what is meant by infinite series, including formula, sum, convergence and worked examples (e.g. geometric and telescoping series). You will understand how to tell if the series is finite or infinite and if it converges or diverges. 0:00 Definition, convergence 1:32 Examples

This video explains what a geometric series is, including formula, example, infinite sum, and rule. The difference between a geometric sequence (e.g. 1,2,4,8,16) and series (e.g. 1+2+4+8+16) that the series is the sum of the terms in the sequence. 0:00 Geometric series, definition, sum 2:09 Worked example

This video explains the nth term test for determining if an infinite series diverges, including, sequence rule, limit formula, conditions, and practice problems (e.g. alternating series). 0:00 nth term test 1:00 Examples

This video explains the integral test for convergence, including the rule, three conditions or requirements for convergence or divergence, and practice problem. 0:00 Integral test definition 1:38 Worked example

This video explains the harmonic series, including definition, how to derive the convergence conditions using the integral test, graph and formula. The harmonic series can be derived from the harmonics or standing waves in physics. 0:00 Harmonic series and p-series 1:33 Convergence conditions 3:37 Worked examples

This video explains the comparison tests for infinite series, including the limit form, rules, practice problems, examples, definition, requirements, what does it say, what is the comp test, conditions (e.g. =0). You will understand when to use the DCT and LCT and what they involve the comparison of. 0:00 Direct comparison test conditions 1:04 Limit comparison test conditions 2:47 Worked examples

This video explains the alternating series test for convergence, including definition and worked example practice problem. The series diverges if any of the three conditions fail. Error bound and remainder are deferred to another tutorial. 0:00 Alternating series test, definition 1:06 Worked example

This video explains the ratio test to determine if an infinite series converges or diverges, including the three conditions and practice problems. 0:00 Ratio test, definition 1:19 Worked examples

This video explains the absolute convergence test of an alternating series, including definition, absolute vs conditional convergence, and what it means in terms of rearranging series that converge absolutely. 0:00 Absolute and conditional convergence 0:59 Rearranging series 1:31 Worked examples