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


b. cos x

Verified step by step guidance

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).
2. 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.
3. Step 3: Calculate f(0). For the function f(x) = cos(x), evaluate f(0) which is cos(0).
4. 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).
5. 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.

Key Concepts

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.