Sample Final

Math 31B

1. The graph of the function f(x) on [0, 1] is given below:

Using this graph:

  1. Estimate

b. Determine how large an error there might be between your estimate and the actual value of this integral. In addition,explain why this error is positive or negative. That is, does your estimate overestimate or underestimate the value of the integral.

Solution

2. a. Show that if n ia a non-negative integer (n >= 0) then

b. Find the volume generated by rotating the curve

around the x-axis.

Solution

3. a. Explain why the function

defined on the domain

has an inverse.

b. Suppose the inverse is called g. Evaluate g'(0).

Solution

4. Let

.

a. How would you define f at 0 (i.e., f(0)) so that f(x) is continuous on the domain x >= 0? Why?

b. Classify the critical points of f(x) on the domain x > 0.

Solution

5. Find the arc length of the curve

.

Solution

6. Evaluate

.

Solution

7. Suppose that a population P = P(t) satisfies the differential equation

where t is measured in years, N = , and P(0) = . Solve the differential eqquation and then determine how long will it take the population to reach 90,000,000

Solution

8. The Taylor expansion of f(x) = ln(1 + x) about a = 0, of order 3, has the form

.

  1. What are

and

?

  1. What is the error in the approximation

Solution

9. Match the differential equation with the direction field:

 

a.

dy/dx = x+y

 

b.

dy/dx = x2

c.

dy/dx = -y

 

d.

dy/dx = y

Solution

 

(The graphs are on the solution)