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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.9.6b
Chapter 3, Problem 3.9.6b

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


b. cos x

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1
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.
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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.
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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.
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