Review for 1st Exam

1. The position of a car is given by the values in the table.
   

   $t$	0	1	2	3	4	5
   $s(t)$	0	10	32	70	119	178

   
   

   Find the average velocity for the time period begining when $t=2$ and lasting
   (a) 3 s; (b) 2 s; (c) 1 s.

2. The displacement of a particle moving in a straight line is given by $s(t)=t^3/6$ . Find the average velocity over the following time period.
   (a) $[1,3]$ ; (b) $[1,2]$ ; (c) $[1,1.5]$ ; (d) $[1,1.1]$ .
3. The point $P(4,2)$ lies on the curve $y=\sqrt{x}$ . If $Q$ is the point $(x,\sqrt{x})$ , find the slope of the secant line $PQ$ (correct to six decimal places) for the following values of $x$ .
   (a) 5; (b) 4.5; (c) 4.1; (d) 4.01; (e) 4.001; (f) 3; (g) 3.5; (h) 3.9; (g) 3.99; (h) 3.999.
4. The point $P(0.5,2)$ lies on the curve $y=1/x$ . If $Q$ is the point $(x,1/x)$ , find the slope of the secant line $PQ$ (correct to six decimal places) for the following values of $x$ .
   (a) 5; (b) 4.5; (c) 4.1; (d) 4.01; (e) 4.001; (f) 3; (g) 3.5; (h) 3.9; (g) 3.99; (h) 3.999.
5. Let $f(x)=3^{x}$ , estimate $f^{\prime}(1)$ from the definition using a small numerical value of $h$ .
6. Let $g(x)=\tan x$ , estimate $g^{\prime}(\pi/4)$ from the definition using a small numerical value of $h$ .
7. Let $f(x)=x^2+3x$ , estimate $f^{\prime}(1)$ from the definition using (a) numerical method; (b) graphic method; (c) algebraic method.
8. Find the equation of the tangent line to $y=3/x$ at $x=3$ .
9. Find the equation of the tangent line to $y=\frac{x}{1-x}$ at $x=0$ .
10. Find the equation of the tangent line to $y=\frac{x}{1+2x}$ at $x=-1/2$ .
11. Problem 23, page 94.
12. Look at the graph of
    $$f(x)= \begin{cases} 0 & x < -\pi/2 \\ \cos x & -\pi/2 < x < \pi/2 \\ 0 & x> \pi/2 \end{cases}$$
    shown in Figure 1. Does $f(x)$ have a derivative at $\pi/2$ ?

    Figure 1: