Download x - Columbia Math - Columbia University

COLUMBIA UNIVERSITY
CALCULUS I (MATH S1101X(3))
SOLUTIONS FOR FINAL EXAM – AUGUST 9, 2012
INSTRUCTOR: DR. SANDRO FUSCO
Problem 1: (10 Points)
Use transformations to sketch the graphs of the functions:
a) [4 points]
y = − sin (2 x )
Answer: 2 pts for setting up problem correctly, 2 pts for correct graph.
b) [3 points]
y = 3 ln( x − 2)
Answer: 2 pts for setting up problem correctly, 1 pt for correct graph.
c) [3 points]
y=
(
1
1+ ex
2
)
Answer: 2 pts for setting up problem correctly, 1 pt for correct graph.
MATH S1101X(3)
FINAL EXAM
PAGE 1 OF 6
Problem 2: (10 Points)
Evaluate the limit if it exists. If the limit does not exist, explain why.
1. [3 points]
 x2 − 9 
 .
lim  2
x→ 3
 x + 2x − 3 
Answer: 2 pts for setting up problem correctly, 1 pt for correct solution.
2. [3 points]
lim+
v→ 4
4−v
.
4−v
Answer: 2 pts for setting up problem correctly, 1 pt for correct solution.
3. [4 points]
lim
x → −∞
x2 − 9
.
2x − 6
Answer: 2 pts for setting up problem correctly, 1 pt for correct value, 1 pt for correct sign.
Problem 3: (10 Points)
f is discontinuous. At which of these numbers is f continuous from
the right, from the left, or neither? Sketch the graph of f .
Find the numbers at which
Answer: 7 pts for setting up problem correctly and correct solution, 3 pts for correct graph.
MATH S1101X(3)
FINAL EXAM
PAGE 2 OF 6
Problem 4: (10 Points)
Compute the derivatives of the following functions:
1. [2 points]
(
y = ln x 2 ⋅ e x
)
Answer: 1 pt for setting up problem correctly, 1 pt for correct solution.
2. [2 points]
(
y = 1 − x −1
)
−1
Answer: 1 pt for setting up problem correctly, 1 pt for correct solution.
3. [3 points]
y = sin x
Answer: 2 pts for setting up problem correctly, 1 pt for correct solution.
4. [3 points]
y = x sin(x )
[Hint: Use logarithmic differentiation]
Answer: 2 pts for setting up problem correctly, 1 pt for correct solution.
Problem 5: (10 Points)
A particle moves along the curve
y = 1 + x 3 . As it reaches the point (2,3), the y-coordinate is
increasing at a rate of 4 cm/s. How fast is the x-coordinate of the point changing at that instant?
Answer: 5 pts for setting up problem correctly, 5 pts for correct solution.
MATH S1101X(3)
FINAL EXAM
PAGE 3 OF 6
Problem 6: (10 Points)
Find two positive integers such that the sum of the first number and four times the second number
is 1,000 and the product of the numbers is as large as possible.
Answer: 5 pts for setting up problem correctly, 5 pts for correct solution.
Problem 7: (10 Points)
f such that: f (0) = 0 , f is continuous and even, f ′( x) = 2 x
if 0 < x < 1 , f ′( x) = −1 if 1 < x < 3 , and f ′( x) = 1 if x > 3 . Justify your answer.
Sketch the graph of the function
Answer: 6 pts for setting up problem correctly, 4 pts for correct graph.
Problem 8: (10 Points)
Compute the following integrals, using any method you like.
∫ (8 x
)
2
1. [2 points]
3
+ 3 x 2 dx
1
Answer: 1 pt for setting up problem correctly, 1 pt for correct solution.
∫ y (y
1
2. [2 points]
2
)
5
+ 1 dy
0
Answer: 1 pt for setting up problem correctly, 1 pt for correct solution.
MATH S1101X(3)
FINAL EXAM
PAGE 4 OF 6
3. [3 points]
∫
e
x
dx
x
Answer: 2 pts for setting up problem correctly, 1 pt for correct solution.
4. [3 points]
∫
x3
dx
1 + x4
Answer: 2 pts for setting up problem correctly, 1 pt for correct solution.
Problem 9: (10 Points)
Find the area of the region bounded by the two parabolas
y = 4 x − x 2 , and y = x 2 .
Answer: 7 pts for setting up problem correctly, 3 pts for correct solution.
Problem 10: (10 Points)
Find the derivatives of the functions below. Explain your answer.
x
a) [4 points]
F ( x) = ∫
0
t2
dt
1+ t3
Answer: 2 pts for setting up problem correctly, 2 pts for correct solution.
MATH S1101X(3)
FINAL EXAM
PAGE 5 OF 6
sin x
b) [6 points]
g ( x) =
∫
1
1- t2
dt
1+ t4
Answer: 4 pts for setting up problem correctly, 2 pts for correct solution.
Problem EC1: (10 Extra Points)
Use the guidelines in Section 4.5 to sketch the curve
y=
1
1 − x2
[A. Domain, B. Intercepts, C. Symmetry, D. Asymptotes, E. Intervals of Increase/Decrease, F. Local
Max/Min Values, G. Concavity and Points of Inflection]
Answer: 6 pts for setting up problem correctly, 4 pts for correct graph.
Enjoy the rest of the summer!
MATH S1101X(3)
FINAL EXAM
PAGE 6 OF 6