Sample Exam
Math 31B
The second hour exam will be given on Tuesday, November 14, from 5 to 6:30. Go to the same classroom you went to for the previous exam.
1 Integrate
Solution:
2. Differentiate
3. Graph y = xln(x) on (0, infinity)
First, set x = 1/n. Then
Second,
Thus, f(x) has a critical point at x = 1/e. Since f(x) = xln(x) is negative for 0 < x < 1, f(1) = 0, and f(x) -> 0 as x -> 0+ , it follows that f has a minimum at x = 1/2. The minimum value is -1/e. The graph will look like this:
4. A piece of roast beef is taken from a refrigerator, where the temperature is 42 degrees (F) and placed in an oven whose temperature is 375. A half hour later the temperature of the meet has risen to 80 degrees. How long will it take for the temperature of the beef to rise to 360 degrees?
Solution. Let T = T(t) be the temperature of the meat at the time t. Then
Numerical Error: the 393s in the last line should be 333s
5. Evaluate
Solution: a).Set x2 = 3u2 Then
b. Set x2 = 2u2. Then
6. Prove that if y = f(x) = tan(x) and g(y) is the inverse of of f(x) then g'(y) = 1/(1+y2)
Starting from g(f(x)) = x we have
Then
Consequently
7. Evaluate
a) First,
Next
So
b. Using L'Hospitals rule:
c. Set x = 1/u. Then
8. Suppose that f(a) = g(a) = 0, f '(x) and g '(x) are continuous and g '(a) is not 0 then
.
NOTE: Do not try to "prove" this by applying L'Hospital's rule. You are being asked to prove L'Hospital's rule. See the top of page 487 of your text for the proof.