Sample Final

Math 31B

1. Evaluate

by the trapezoidal rule, using n = 4.

answer: Trapezoidal sum = 1.00409

2. A sphere is of radius 2 foot, and full of water. How much work is done in pumping all the water in the tank to a point one foot above the top of the sphere?

answer:

3. Let

a) Explain why f(x) has an inverse function g(y) on

Since f'(x) = (e^x +4e^-x)/2 > 0 for all x, f(x) is an increasing function of x, so is one to one on its domain, so it has an inverse there.

b) Prove that

Starting from y = f(x) and and differentiating with respect to x we get

.

So the problem is one of expressing f'(x) in terms of y. To this end:

That is,

.

Since f'(x) > 0 for all x it follows that

Consequently,

4. Evaluate

Hint in (a) let

a) Trying to evaluate (a) by L'Hospital's rule leads to a mushrooming explosion. So, try a substitution:

So

b) The limit is 0. Make the substitution x = 1/n and then take the limit as n -> infinity

5. Let

Evaluate f'(5).

This is a problem in logarithmic differentiation:

So

6. Integrate

7.

Integrate by parts:

8. Approximate sin(x) using a Taylor expansion of order 4, expanded around a = 0 for x = 1/4. Give an upper bound for the error in this approximation.

Solution:

Consequently

9.Integrate

10. Integrate

11. Find the area of the region that is bounded above by the circle with center at (0, 3/4) and radius 1/2 and is bounded below by the circle with center at (0,0) and radius 1.

12. Evaluate

a) b)

13. Let A be the area region bounded by the curve y = 1/x², and B be the region bounded by the curve y = ln(x)/x², . Is A < B,

A > B or A = B? Why?

A = B; they are both equal to 1

14. Find the solution of the differential equation

solution: u² = t² + tan(t) + 25

15. A tank whose capacity is 2000 gallons contains 1000 gallons of pure water. A salt solution containing .3 pounds of salt per gallon of water

is pumped in at the rate of 2 gallons a minute, is thoroughly mixed, and is pumped out at the rate of 2 gallon a minute. What is the concentration of salt per gallon of mix at the time t? How much salt is in the tank "after a long time"?

solution: A(t) = total amount of salt in the tank at time t

A(t)= 300(1-exp(-(1/500)t)

A(t) -> 300 as t -> infinity