Practice Problems

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

(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?

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

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 $(a)-(d)$ :

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

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 $(3,6)$ ?