Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).

Verified step by step guidance

1. First, understand the concept of linear approximation. Linear approximation is a method used to approximate the value of a function near a given point using the tangent line at that point.
2. For a function f(x), the linear approximation at a point x = a is given by the formula L(x) = f(a) + f'(a)(x - a), where f'(a) is the derivative of f at x = a.
3. Consider the function f(x) = mx + b, which is a linear function. The derivative of f(x) with respect to x is f'(x) = m, since the derivative of mx is m and the derivative of a constant b is 0.
4. Substitute f(x) = mx + b and f'(x) = m into the linear approximation formula: L(x) = f(a) + m(x - a). Since f(a) = ma + b, the formula becomes L(x) = ma + b + m(x - a).
5. Simplify the expression: L(x) = ma + b + mx - ma = mx + b. Notice that L(x) = f(x), which confirms that for a linear function, the linear approximation at any point is indeed the function itself.

Key Concepts

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. This function produces a straight line when graphed, and its properties include constant rate of change and predictable behavior across its domain.

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. For a function f at point a, the linear approximation L(x) is given by L(x) = f(a) + f'(a)(x - a). This technique is particularly useful for simplifying complex functions in calculus.

Tangent Line

The tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. In the context of linear approximation, the tangent line serves as the best linear estimate of the function's behavior near that point, making it crucial for understanding how well the approximation represents the function.