2. Intro to Derivatives

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

Based on the graph of $f\(\left\)(x\(\right\))$, describe the graph of the derivative $f^{\(\prime\)}\(\left\)(x\(\right\))$ on the interval $\(\left\)(0,\(\infty\]\right\))$.

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Recovering a function from its derivative

b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. <IMAGE>

a. At what time is the rate of change of the charge Q' the greatest?