Math 101—April 2015 Final Examination
Duration: 150 minutes
Problems 1–2 are answer-only questions: the correct answers in the given boxes earns full credit,
and no partial credit is given.
1a. [2 pts] Only one of the following statements is always true; determine which one is true.
(Assume f (x) and g(x) are continuous functions.)
Answer:
Z −2
Z 2
A:
f (x)dx = −
f (x)dx.
−3
3
Z 1
Z 1
Z 1
B:
f (x) · g(x) dx =
f (x) dx ·
g(x) dx.
0Z
0
0
Z ∞
∞
C: If
f (x) dx converges and g(x) ≥ f (x) ≥ 0 for all x, then
g(x) dx converges.
1
D: If
∞
X
1
(−1)n+1 bn converges, then
n=1
∞
X
bn also converges.
n=1
E: When f (x) is positive and concave up, any Trapezoid Rule approximation for
Z b
will be an upper estimate for
f (x) dx.
a
Z
b
f (x) dx
a
1b. [2 pts] For each integral, choose the substitution type that is most beneficial for evaluating
the integral. (Write F, G, or H in each box; each answer will be used exactly once.)
F: x = a sec θ
G: x = a sin θ
H: x = a tan θ
Z
2x2
√
dx
Answer:
(i)
2 − 16
9x
Z
x4 − 3
√
(ii)
dx
Answer:
2
1
−
4x
Z
(iii)
(25 + x2 )−5/2 dx
Answer:
1c. [2 pts] For which values of the constant q does the integral
Z
1
xq dx converge?
0
Answer:
It converges exactly when q < −1.
It converges exactly when q ≤ −1.
It converges exactly when q > −1.
It converges exactly when q ≥ −1.
It converges for all q.
Z 7
1d. [2 pts] Calculate
x3 + sin(2x) + 1 dx. Simplify your answer completely.
J:
K:
L:
M:
P:
−7
Answer:
Math 101—April 2015 Final Examination
page 2
Problems 1–2 are answer-only questions: the correct answers in the given boxes earns full credit,
and no partial credit is given.
Z 2
2a. [2 pts] Which of the following integrals equals
f (x3 ) dx?
Answer:
1
A:
B:
C:
Z
Z
Z
2
2
3u f (u) du
D:
1
Z
2
1
√
3
2
2
3u f (u) du
E:
1
Z
√
3
1
8
2
3u f (u) du
F:
1
Z
f (u)
du
3u2/3
8
1
G:
2b. [2 pts] Which of the following Maclaurin series equals
f (u) du
1
2
f (u)
du
3u2/3
f (u)
du
3u2/3
2
Z
H:
√
3
Z
2
f (u) du
1
J:
Z
8
f (u) du
1
x4
?
1 − 2x3
Answer:
K : x3 − 2x7 + 4x11 − 8x15 + · · ·
Q : x4 − 2x7 + 4x10 − 8x13 + · · ·
M : x3 + 2x7 + 4x11 + 8x15 + · · ·
T : x4 + 2x7 + 4x10 + 8x13 + · · ·
L : x3 − 21 x7 + 41 x11 − 81 x15 + · · ·
P : x3 + 12 x7 + 41 x11 + 18 x15 + · · ·
S : x4 − 12 x7 + 41 x10 − 81 x13 + · · ·
U : x4 + 21 x7 + 14 x10 + 18 x13 + · · ·
2c. [4 pts] For each of the following series, choose the appropriate statement. (Write the
appropriate two-letter answer in each box; each answer will be used at most once.)
R∞
CI: Converges by Integral Test with 1 lnxx dx
CT: Converges by Test for Divergence P
1
CC: Converges by Comparison Test with ∞
n=1 n3
CA: Converges by Alternating Series Test
P∞ 1
CL: Converges by Limit Comparison
Test
with
n=1 n2
R ∞ ln x
DI: Diverges by Integral Test with 1 x dx
DT: Diverges by Test for Divergence P
√1
DC: Diverges by Comparison Test with ∞
n=1 n
DA: Diverges by Alternating Series Test
P
π
DL: Diverges by Limit Comparison Test with ∞
n=1 n2
(i)
∞
X
ln n
n=1
(ii)
∞
X
n=1
(iii)
(iv)
∞
X
n=1
∞
X
n=1
n
Answer:
4n − 1
(2n2 + 1)2
Answer:
(−1)n n
Answer:
arctan n
n2
Answer:
Math 101—April 2015 Final Examination
page 3
Problems 3–6 are short-answer questions: put a box around your final answer, but you must include
the correct accompanying work to receive full credit.
√
3a. [3 pts] CalculateZthe area of the region enclosed by y = 2x and y = x + 1.
3b. [3 pts] Evaluate
tan3 x sec5 x dx.
4a. [3 pts] Find the average
value of the function y = x2 ln x on the interval 1 ≤ x ≤ e.
Z
1
dx.
4b. [3 pts] Calculate
x4 + x2
∞
X
(−1)n
is absolutely convergent, conditionally
5a. [3 pts] Determine whether the series
9n + 5
n=1
convergent, or divergent; justify your answer.
∞
X
(−1)n+1
5b. [3 pts] The series
converges to some number S (you don’t have to prove
(2n + 1)2
n=1
this). According to the Alternating Series Estimation Theorem, what is the smallest value
1
away from S? For this value
of n for which the nth partial sum of the series is at most 100
of n, write out the nth partial sum of the series.
6a. [3 pts] Find the Maclaurin series for
Z
x4 arctan(2x) dx.
6b. [3 pts] Suppose that you have a sequence {bn } such that the series
∞
X
n=0
(1 − bn ) converges.
Using the tests we’ve learned in class, prove that the radius of convergence of the power
∞
X
bn xn is equal to 1.
series
n=0
Problems 7–11 are long-answer: give complete arguments and explanations for all your calculations—
answers without justifications will not be marked.
Z x3 +1
3
3
7. Define f (x) = x
et dt.
0
(a) [3 pts] Find a formula for the derivative f ′ (x).
(b) [3 pts] Find the equation of the tangent line to the graph of y = f (x) at x = −1. Simplify
your answer completely.
√
8. Let D be the region below the graph of the curve y = 9 − 4x2 and above the x-axis.
(a) [4 pts] Using an appropriate integral, find the area of the region D; simplify your answer
completely.
(b) [3 pts] Find the centre of mass of the region D; simplify your answer completely. (Assume
it has constant density ρ.)
Math 101—April 2015 Final Examination
page 4
9.
(a) [3 pts] Find the solution of the differential equation
1+
p
y 2 − 4 ′ sec x
y =
that satisfies
tan x
y
y(0) = 2. You don’t have to solve for y in terms of x.
(b) [4 pts] A sculpture, shaped like a pyramid 3 m high sitting on the ground, has been made
by stacking smaller and smaller iron plates on top of one another. The iron plate at height
z m above ground level is a square whose side length is (3 − z) m. All of the iron plates
started on the floor of a basement 2 m below ground level.
Write down an integral that represents the work, in newtons, it took to move all of the
iron from its starting position to its present position. Do not evaluate the integral. (You can
use 9.8 m/s2 for the force of gravity and 8000 kg/m3 for the density of iron.)
10. Let R be the region inside the circle x2 + (y − 2)2 = 1. Let S be the solid obtained by rotating
R about the x-axis.
(a) [4 pts] Write down an integral representing the volume of S.
(b) [3 pts] Evaluate the integral you wrote down in part (a). (You may use any technique you
know, including a geometric one, to do so.)
11.
(a) [4 pts] Find the value of the convergent series
∞ n+1
X
2
1
1
+
−
3n
2n − 1 2n + 1
n=2
(simplify your answer completely).
(b) [4 pts] Find the interval of convergence for the power series
∞
X
(x − 2)n
.
4/5 (5n − 4)
n
n=1