Exam 1

1. (10%) The position of a car is given by the values in the table.
   

   $t$	0	1	2	3	4	5
   $y$	1.8	27.4	43.3	49.4	45.7	32.3

   
   

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

2. (10%) The displacement of a particle moving in a straight line is given by $y=80-4.9t^{2}$ . Find the average velocity over the following time period.
   (a) $[1,2]$ ; (b) $[1,1.1]$ .
3. (10%) 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) 4.5; (b) 4.001.
4. (10%) Let $g(x)=\tan x$ , estimate $g^{\prime}(\pi/4)$ from the definition using a small numerical value of $h$ .
5. (20%) Let $f(x)=5x^2$ , estimate $f^{\prime}(10)$ from the definition using (a) numerical method; (b) algebraic method.
6. (15%) Find the equation of the tangent line to $y=x^{3}$ at $x=-2$ .

7. (20%) The graph of a function $f(x)$ is shown in Figure 2.

   Figure 2:

   
   
   
   Indicate whether $f$ , $f^{\prime}$ , $f^{\prime\prime}$ at each marked point is positive, negative, or zero.
   

   point	$A$	$B$	$C$	$D$	$E$
   $f$
   $f^{\prime}$
   $f^{\prime\prime}$

   
   
8. (5%) Look at the graph of
   $$f(x)= \begin{cases} 0 & x < -\pi/2 \\ \sin 2 x & -\pi/2 < x < \pi/2 \\ 0 & x> \pi/2 \end{cases}$$
   shown in Figure 3. Does $f(x)$ have a derivative at $\pi/2$ ?

   Figure 3: