MAT 126 Calculus B

Midterm 1

February 21, 2000

SHOW ALL YOUR WORK ON THESE PAGES! TOTAL SCORE = 100

1.

(30 points)

 

Figure: The graph of f(x) .

The graph of a function f(x) is shown in Figure 1.

(a)

Estimate the area under the graph between x=2 and x=7 .

(b)

Estimate ${\displaystyle \int_0^7 f(x)~dx}$

2.

(20 points) A car is traveling back and forth along a straight road that runs East-West. We make the convention that positive velocity means velocity Eastward. During a half hour, we measure the velocity (in miles per hour) every 6 minutes (6 minutes = 0.1 hour), with the following results.

$$\begin{array}{c\vert c\vert c\vert c\vert c\vert c\vert c} \p... ...velocity~in~m.p.h} & 20 & 30 & 20 & -10 & -10 & 0 \end{array}$$

Use the data in the table to estimate:

(a)

Where the car is at the end of the half hour with respect to where it was at the beginning. Tell us explicitly how you are making the estimate!

(b)

How far the car drove (total mileage) during the half hour. Tell us explicitly how you are making the estimate!

3.

(30 points) Find anti-derivatives of the following functions:

(a)

$f(x)= x^2 - 2\cos(x)$

(b)

$f(x)= (1/2)\sin(3x)$

(c)

$f(x)= \frac{\textstyle 5}{\textstyle \sqrt{x}}$

4.

(20 points) Calculate the following definite integrals using the ``Evaluation Theorem.'' You must use the Evaluation Theorem to get credit!

(a)

${\displaystyle \int_0^2 x^5~dx}$

(b)

${\displaystyle \int_{-1}^1 \frac{\textstyle 3}{\textstyle 1+x^2}~dx}$