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. Understand that the linear approximation of a function f at a point a is given by L(x) = f(a) + f'(a)(x - a). This means we need to find the value of the function and its derivative at the point a.
2. For the function f(x) = mx + b, calculate the derivative f'(x) which is constant and equal to m, since the slope of a linear function is constant.
3. Choose a specific point a to evaluate the linear approximation. For example, let a = 0. Then, f(0) = b and f'(0) = m.
4. Substitute these values into the linear approximation formula: L(x) = b + m(x - 0) = mx + b, which simplifies to L(x) = f(x).
5. Conclude that the statement is true because the linear approximation L(x) at any point a for the linear function f(x) = mx + b is indeed equal to f(x).