Chapter 2, Problem 2RE

In parts (a)-(b), use the function $y=\frac{1}{2}{x}^2$ .

(a) Find the average rate of change of $y$ with respect to $x$ over the interval $\left[3,4\right]$ .

(b) Find the instantaneous rate of change of $y$ with respect to $x$ at $x=3$ .

(c) Find the instantaneous rate of change of $y$ with respect to $x$ at a general $x\text{-value}\text{.}$

(d) Sketch the graph of $y=\frac{1}{2}{x}^2$ together with the secant line whose slope is given by the result in part (a), and indicate graphically the slope of the tangent line that corresponds to the result in part (b).