Math 180 – Spring 2012
Name:
First Exam
DO NOT WRITE ON THIS PROBLEM SHEET
Nothing written here will be read or graded.
WRITE YOUR ANSWERS IN AN EXAM BOOKLET
(20 pts) 1. Let c be a real number. Given
f (x) =
(x − 1)2 x ≤ 3
6 − cx
x >3
answer the following questions:
(a) Find the left hand limit, limx→3− f (x).
(b) Find the right hand limit, limx→3+ f (x).
(c) Find the value of c for which the ordinary limit limx→3 f (x) exists.
(d) Is f (x) continuous at x = 2?
(20 pts) 2. Find each limit or explain why it does not exist.
3x 2 + 4
.
x→0 x 2 + 6x + 1
x 2 + 3x + 2
(b) lim
.
x→−2 2x(x + 2)
x 10 − 2x 3 + 5
(c) lim
.
x→∞
x2 + 1
(a) lim
(30 pts) 3. Compute the derivative of each function below. Leave your answer in an unsimplified form
so it is clear what method you used.
(a) x 2 sin(x)
e x − e −x
(b)
1+x
√
(c) tan( x)
(10 pts) 4. Using the Intermediate Value Theorem show that the equation x 3 = 3x + 1 has a solution.
(20 pts) 5. Given the function
f (x) = x 2 + 1
answer the following questions:
(a) Using the definition of the derivative as a limit compute f 0 (1).
(b) Give the equation of the tangent line to the graph of f at the point (1, 2).
Hand in this problem sheet along with your exam booklet!