Sample Exam

Math 31B

1. Two functions y = f(x) and y = g(x) are sketched below. The curve for f(x) always lies above that for g(x).

Which of the four numbers below is the best estimate for

?

a) 5.5 b) 6.75 c) 8.25 d)9

answer: 6.75

2. Let f(x) be the curve graphed below on [0,1]

and R be the region bounded by f(x) and the line y = 1/2. Write down

the integral that represents the volume generated when R is

a) Rotated around the line y = 1/2

b) Rotated around the line y = 0

c) Rotated around the line x = 0

d) Rotated around the line x = 2

Just write down the integral; you do not have to evaluate it.

a)

b)

c)

d)

 

3. The curve f(x) = x3 + x , for x in [0, 2], is graphed below.

Let g(y) be the inverse of f(x). Evaluate

From the graph

Then

So

 

4. Graph the function f(x) = (x2 +1)ex on the interval [-4,0], showing all local minimums and maximums, and its concavity. Be sure to label all these points.

Note first that

Next

That is

so f(x) is always increasing on its domain.

As for the second derivative,

so the graph has a change of conacavity at x = -3 and x = -1. Furthermore, the second derivative is negative for x in (-3,-1) so

the graph is concave down in this interval.

The graph can be sketched, free-hand, from these facts.

5. A bowl has the shape of a semi-sphere of radius 2 foot, and is filled with water. How much work is done in pumping out the top foot of water in the bowl? The water is pumped to the top of the bowl. Take the weight of water to be 62.5 lbs a cubic foot.

 

A typical slab, lying y units above the x-axis, has volume Pi x2 dy and weight 62.5 Pi x2 dy . It is lifted a height of (2-y) foot. So the work done is given by the integral

 

6. Let f(x) = x2 +ax + 3, where a is a positive constant. Suppose that the average value of f(x) on [0, a] is 19/30. What is the value of a?

ans sqrt(12)

7. a. Find the derivative of f(x) = (x-1)ex .

b. Using part (a) find the volume of the solid generated by

rotating the region bounded by y = ex , 0 <= x <= 2, the

line y = 0, and the line x = 2 around the y-axis.

a) We have

This says that the anti-derivative of xex is (x-1)ex.

b)Using shells, the volume is