COLUMBIA UNIVERSITY
CALCULUS I (MATH S1101X(3))
FINAL EXAM – AUGUST 9, 2012
INSTRUCTOR: DR. SANDRO FUSCO
FAMILY NAME: ______________________________________________________________
GIVEN NAME: ______________________________________________________________
INSTRUCTIONS:
1.
2.
3.
4.
5.
6.
7.
You have 3 hours.
Your work must justify the answer you give.
There are 10 questions (Problems 1-10) plus 1 bonus question (Problem EC1).
Attempt the first ten questions first, and then work on the bonus question.
No calculators, lecture notes and/or books are permitted.
Point values are as shown.
This is the first of thirteen (13) pages.
Question
Topic
Points
1
Functions
10
2
Computing Limits
10
3
Continuity
10
4
Computing Derivatives
10
5
Implicit Differentiation / Related Rates
10
6
Optimization Problems
10
7
Curve Sketching / Antiderivatives
10
8
Computing Integrals
10
9
Area Between Curves
10
10
Fundamental Theorem of Calculus
10
Total
EC1
MATH S1101X(3)
Marks
100
Extra Credit
FINAL EXAM
10
PAGE 1 OF 13
Problem 1: (10 Points)
Use transformations to sketch the graphs of the functions:
a) [4 points]
y = − sin (2 x )
b) [3 points]
y = 3 ln( x − 2)
c) [3 points]
y=
MATH S1101X(3)
(
1
1+ ex
2
)
FINAL EXAM
PAGE 2 OF 13
Problem 2: (10 Points)
Evaluate the limit if it exists. If the limit does not exist, explain why.
1. [3 points]
2. [3 points]
3. [4 points]
MATH S1101X(3)
 x2 − 9 
 .
lim  2
x→ 3
x
+
2
x
−
3


lim
v→ 4+
lim
x → −∞
4−v
.
4−v
x2 − 9
.
2x − 6
FINAL EXAM
PAGE 3 OF 13
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
1 + x2

f (x) = 2 − x

2
(x − 2)
MATH S1101X(3)
FINAL EXAM
if x ≤ 0
if 0 < x ≤ 2
if x > 2
PAGE 4 OF 13
Problem 4: (10 Points)
Compute the derivatives of the following functions:
(
1. [2 points]
y = ln x 2 ⋅ e x
2. [2 points]
y = 1 − x −1
3. [3 points]
y = sin x
4. [3 points]
y = x sin(x )
MATH S1101X(3)
(
)
)
−1
[Hint: Use logarithmic differentiation]
FINAL EXAM
PAGE 5 OF 13
Problem 5: (10 Points)
A particle moves along the curve y = 1 + x . 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?
3
MATH S1101X(3)
FINAL EXAM
PAGE 6 OF 13
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.
MATH S1101X(3)
FINAL EXAM
PAGE 7 OF 13
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
MATH S1101X(3)
FINAL EXAM
PAGE 8 OF 13
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
∫ y (y
1
2. [2 points]
2
)
5
+ 1 dy
0
MATH S1101X(3)
FINAL EXAM
PAGE 9 OF 13
3. [3 points]
4. [3 points]
MATH S1101X(3)
x
∫
e
∫
x3
dx
1 + x4
x
dx
FINAL EXAM
PAGE 10 OF 13
Problem 9: (10 Points)
Find the area of the region bounded by the two parabolas
MATH S1101X(3)
FINAL EXAM
y = 4 x − x 2 , and y = x 2 .
PAGE 11 OF 13
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
sin x
b) [6 points]
g ( x) =
∫
1
MATH S1101X(3)
1- t2
dt
1+ t4
FINAL EXAM
PAGE 12 OF 13
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]
Enjoy the rest of the summer!
MATH S1101X(3)
FINAL EXAM
PAGE 13 OF 13