## Practice Problems

Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.

1. Use the function in Fig. 12 to fill in the table and then graph $m(x)$.

$x$  $y = f(x)$ $m(x)$ = the estimated slope of the tangent line to $y=f(x)$ at the point (x,y)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

3.

(a) At what values of $x$ does the graph of f in Fig. 14 have a horizontal tangent line?

(b) At what value(s) of $x$ is the value of f the largest? smallest?

(c) Sketch the graph of $m(x)$ = the slope of the line tangent to the graph  of $f$ at the point $(x,y)$

5.

(a) Sketch the graph of $f(x) = sin(x) for –3 ≤ x ≤ 10$

(b) Sketch the graph of $m(x)$ = slope of the line tangent to the graph of $sin(x)$ at the point $(x, sin(x))$

(c) Your graph in part (b) should look familiar. What function is it?

Problems 7 – 9 assume that a rocket is following the path $y = x^2$, from left to right.

7. At what point should the engine be turned off in order to coast along the tangent line to a base at $(5,16)$

9. At what point should the engine be turned off in order to coast along the tangent line to a base at $(1,3)$

For each function $f(x)$ in problems 11-15, perform steps $(a)-(d)$:

(a) calculate $\mathrm{m}_{\mathrm{sec}}=\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})}{(\mathrm{x}+\mathrm{h})-(\mathrm{x})}$ and simplify

(b) determine $\mathrm{m}_{\mathrm{tan}}=\lim _{h \rightarrow 0} \mathrm{~m}_{\mathrm{Sec}}$

(c) evaluate $\mathrm{m}_{\tan }$ at $\mathrm{x}=2$

(d) find the equation of the line tangent to the graph of $\mathrm{f}$ at $(2, \mathrm{f}(2))$

11. $f(x)=3 x-7$

13. $\mathrm{f}(\mathrm{x})=\mathrm{ax}+\mathrm{b}$ where $\mathrm{a}$ and $\mathrm{b}$ are constants

15. $f(x)=8-3 x^{2}$

In problem 17, use the result that if $f(x)=a x^{2}+b x+c$ then $m_{t a n}=2 a x+b$.

17. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+2 \mathrm{x} .$ At which point(s) $(\mathrm{p}, \mathrm{f}(\mathrm{p}))$ does the line tangent to the graph at that point also go through the point $(3,6)$?