EXAM
Exam 1
Math 1331, Summer 2010
July 20, 2010
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 7 problems. There are 370 points
total.
Good luck!
40 pts.
Problem 1. In each part, carry out the indicated operations and simplify.
A.
(2x + 3)2
B.
(2x − 1)(3x + 5)
C.
(x − 1)4 − (x − 1)2 x2
(x − 1)5
D.
60 pts.
1
2
x2
− 2+
x x
x+2
Problem 2. According to company reports, Starbucks’ annual sales (in billions of dollars) for 1998 through 2003 are as follows
Year
Sales
1998
1.28
1999
1.73
2000
2.18
2001
2.65
2002
3.29
2003
4.08
A. Find an equation for the least squares line for this data (let x = 0 represent
the year 1998). Use a calculator.
B. What was the estimated rate of change in Starbucks’ annual sales over this
period?
C. If the trend continued, what were the estimated sales in 2006?
60 pts.
Problem 3. Each part shows the graph of a function f (x). In each part,
answer the following questions:
1. Find
lim f (x).
x→2−
2. Find
lim f (x).
x→2+
3. Find
lim f (x).
x→2
4. Find f (2).
5. Is the function continuous at x = 1? Explain your answer.
1
A.
y
4
3
2
1
−1
0
0
1
−1
2
2
3
x
B.
y
4
3
2
1
−1
0
0
1
−1
3
2
3
x
C.
y
4
3
2
1
−1
0
0
1
2
3
x
−1
60 pts.
Problem 4. In each part, find the limit (“Does Not Exist” is a possible answer).
A.
x2 + x + 1
x→3
x−1
lim
B.
lim
x→2
C.
1
(x − 2)2
x2 − 5x + 6
x→2
x−2
lim
40 pts.
Problem 5. Find the equation of the tangent line to the graph of f (x) =
x2 + 3x − 1 at the point x = 2.
4
Problem 6. In each part, find the derivative f 0 (x) for the given function f (x).
70 pts.
A.
f (x) = 6x5 + 3x4 − 5x3 + 2x2 − x + 17.
B.
√
2
5
2
f (x) = √ + 3 x + − 5
x x
x
C.
f (x) = (3x3 − 2x2 + 5x + 1)(7x3 + 8x2 + x + 12)
D.
f (x) =
2x + 1
x2 + 1
E.
f (x) = (3x2 + 2x + 5)6
F.
x
(x2 + 1)2
f (x) =
G.
f (x) = √
40 pts.
3x2
1
+ 2x − 3
Problem 7. In each part, find the derivative f 0 (x) for the given function
Suppose that f (x) such that f 0 (x) exits for all x, and that the following
conditions are satisfied:
f 0 (0) = 5
f (0) = −7
f 0 (1) = 3
f (1) = −10
f 0 (2) = −3
f (2) = 0.
Let g(x) be defined by
g(x) = f (x2 + 2x + 1).
Use the chain rule for find g 0 (0).
5