Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Linearization

Calculus

4. Applications of Derivatives

Linearization

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2:10 minutes

Problem 3.9.6b

Textbook Question

Common linear approximations at x = 0 Find the linearizations of the following functions at x = 0.

b. cos x

Verified step by step guidance

Step 1: Understand the concept of linearization. Linearization is the process of approximating a function by a line near a given point. For a function f(x), the linearization at x = a is given by L(x) = f(a) + f'(a)(x - a).

Step 2: Identify the function and the point of approximation. Here, the function is f(x) = cos(x) and the point of approximation is x = 0.

Step 3: Calculate f(0). For the function f(x) = cos(x), evaluate f(0) which is cos(0).

Step 4: Find the derivative of the function, f'(x). The derivative of cos(x) is -sin(x). Evaluate f'(0) which is -sin(0).

Step 5: Substitute f(0) and f'(0) into the linearization formula L(x) = f(0) + f'(0)(x - 0) to find the linear approximation of cos(x) at x = 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linearization

Linearization is the process of approximating a function near a given point using the tangent line at that point. For a function f(x) at x = a, the linearization is given by L(x) = f(a) + f'(a)(x - a). This provides a simple way to estimate function values near a, especially when the function is complex.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. For the cosine function, the derivative is -sin(x). At x = 0, this derivative helps determine the slope of the tangent line, which is crucial for finding the linear approximation.

Cosine Function

The cosine function, cos(x), is a periodic function that describes the x-coordinate of a point on the unit circle as the angle x varies. At x = 0, cos(x) equals 1. Understanding the behavior of cos(x) around x = 0 is essential for accurately applying linearization techniques to approximate its values.