Chapter 11.3, Problem 75E

a.

To determine

To explain: how can use a secant line to approximate the tangent line at ( x, f(x))

Explanation of Solution

Given information: Given graph of the function f.

Calculation:

The secant line is a line through the point of tangency ( x, f(x)) and another point

( x + h, f(x + h)) on the graph of f . As the second point “moves” closer to the point of tangency, $h\to 0$ , and the slope of the secant line become a better approximation to that of the slope of the tangent line at the point of tangency.

b.

To determine

To explain: how can use the limit process to find the exact slope of the tangent line at ( x, f(x)).

Explanation of Solution

Given information: Given graph of the function f.

Calculation:

Thus the limit of the slop of the secant line becomes the slope of the tangent line at the point of tangency as h approaches 0: $\underset{h\to \infty }{\lim }\frac{f(x+h)-f(x)}{h}$ .