Chapter 2.1, Problem 12ES

A function $y=f\left(x\right)$ and values of $x_0$ and $x_1$ are given.

(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[x_0,x_1\right]$ .

(b) Find the instantaneous rate of change of $y$ with respect to $x$ at the specified value of $x_0$ .

(c) Find the instantaneous rate of change of $y$ with respect to x at an arbitrary value of $x_0$ .

(d) The average rate of change in part (a) is the slope of a certain secant line, and the instantaneous rate of change in part (b) is the slope of a certain tangent line. Sketch the graph of $y=f\left(x\right)$ together with those two lines.

$y=2x^3;x_0=1,x_1=2$