Скачать или смотреть Linear Approximation and Differentials in Calculus | Tangent Line Approximations

📚 Learn How to Use Linear Approximations and Differentials
In this video, you’ll learn how to use tangent lines to approximate complicated functions and apply differentials to estimate error. We’ll cover the linearization formula, work through examples, and see how to apply these techniques to real-world problems.

🔍 What You’ll Learn in This Video:
✅ Linearization Formula
✔ L(x) = f(a) + f′(a)(x – a)
✔ How tangent lines approximate functions near x = a
✔ Examples with polynomials and trig functions

✅ Accuracy of Linear Approximations
✔ Absolute error: E(x) = |f(x) – L(x)|
✔ How to determine when an approximation is within a desired tolerance

✅ Differentials
✔ dy = f′(x) dx as an estimate for Δy
✔ How to use differentials for quick calculations
✔ Example: y = √x, x = 1, dx = 1

✅ Applications and Estimations
✔ Using linear approximation to estimate cos(31.5°)
✔ Estimating error in the area of a circle when radius measurements have uncertainty

By the end of this video, you’ll understand how linear approximations and differentials work, how accurate they are, and how to apply them to estimate values and errors in calculus.

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Timestamps
00:00 Introduction
00:45 Linearization Formula and Differentials
04:46 Examples - Finding a Linearization
09:53 Example - Determining the Accuracy of a Linear Approximation
16:39 Example - Finding a Differential
20:26 Example - Linear Approximation and Differentials for Estimation
27:48 Example - Estimating Error in Application Problems

More About Understand The Math
Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples.

Each video includes free guided notes, linked in the description. These notes help students follow along, highlight key concepts, and practice problems, making review easier and more effective.

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