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Lennie Euler
Calculus II — sample problems
�
3. (1 pt) Evaluate the definite integral.
1) sinn (a x) · cos(a x) dx
� e3
�
ax−b
2)
dx with B = 2 b
a x2 − B x + c
� en
dx
3)
1 x · (1 + ln(x))
4) compute area between 2 curves: integrate wrt x, then wrt y
5) solid of revolution: rotate about y = H
6) work
to lift a leaking bucket
�
7) � ea x · sin(b x) dx
8) a x · ln(b x) dx
1
4. (1 pt) Consider the area between the graphs x + 1y = −2
and x + 8 = y2 . This area can be computed in two different ways
using integrals.
First of all it can be computed as a sum of two integrals
� b
a
�
9) xa · cos(xb ) dx with a = 2 b − 1
cubic
x 2 − a2
quadratic
11) partial fractions for
with x − a being a
(x − a) · (quadratic)
factor of the denominator quadratic
13)
�
� ∞
a
dx
√
dx
2
x · x2 + a2
with the following values:
α=
β=
h(y) =
b
g(x) dx
h(y) dy
Either way we find that the area is
y=
22) Do you believe Maclaurin series for cosine is a function?
23) Taylor polynomial for x3 centered at A
24) match: graphs of curves, parametric equations
a
25) Cartesian equation for polar curve: r =
b cos(θ) + c cos(θ)
1.
� (1 pt) Evaluate the indefinite integral.
+C.
2.
� (1 pt) Evaluate the indefinite integral.
3x − 2
dx =
2
(3x − 4x + 5)6
+C.
.
5. (1 pt) Find the volume of the solid obtained by rotating the
region bounded by the given curves about the the line y = −3.
a
; radius
1 − x3
21) coefficients c0 , . . . , c4 of Taylor series centered at A for
ln(x); interval of convergence
sin7 (4x) cos(4x) dx =
� β
α
14) match: sequence, convergence/divergence behavior
15) geometric series: evaluate if convergent
16) match: series, convergence/divergence behavior
17) 2-phase match: series, behavior, test technique
18, 19) convergence interval for a power series
of convergence
� c
Alternatively this area can be computed as a single integral
x · eb x dx if it converges (b is negative)
20) coefficients c0 , . . . , c4 of Maclaurin series for
f (x) dx +
with the following values:
a=
b=
c=
f (x) =
g(x) =
10) partial fractions for
12) evaluate
dx
x(1 + ln x)
1
, y = 0, x = 2, x = 7
x4
Answer:
6. (1 pt) A bucket that weighs 3.2 pounds and a rope of negligible weight are used to draw water from a well that is 83 feet
deep. The bucket is filled with 38 pounds of water and is pulled
up at a rate of 2.7 feet per second, but water leaks out of a hole
in the bucket at a rate of 0.15 pounds per second. Find the work
done pulling the bucket to the top of the well. Your answer must
include the correct units. (You may enter lbf or lb*ft for ft-lb .)
Work =
1
(Show the student hint after 1 attempts: )
14. (1 pt) Match each sequence below to statement that
BEST fits it.
Hint: Calculate the work done lifting the bucket (without
the water) and the water (without the bucket) separately and add
them to find the total work done. Find a function that describes
the weight of the water in the bucket when the bucket is x feet
above its starting position.
7.
� (1 pt) Evaluate the indefinite integral.
e4x sin(6x) dx =
8.
� (1 pt) Evaluate the indefinite integral.
5x ln(4x) dx =
9.
the indefinite integral.
� (1 pt) �Evaluate
�
5
3
x cos x dx =
STATEMENTS
Z. The sequence converges to zero;
I. The sequence diverges to infinity;
F. The sequence has a finite non-zero limit;
D. The sequence diverges.
+C.
SEQUENCES
1. sin(n)
n100
2. (1.01)
n
3. ln(ln(ln(n)))
3 −5n
4. n3n−n
5
n!
5. n1000
+C.
+C.
Hint: First make a substitution and then use integration by
parts to evaluate the integral.
6. (ln(n))
n
7. n sin ( 1n )
8. arctan(n + 1)
10. (1 pt) Note: You can get full credit for this problem by
just entering the final answer (to the last question) correctly. The
initial questions are meant as hints towards the final answer and
also allow you the opportunity to get partial credit.
�
4x3 + 3x2 − 33x − 30
Consider the indefinite integral
dx
x2 − 9
15. (1 pt) The following series are geometric series.
Determine whether each series converges or not.
For the series which converge, enter the sum of the series. For
the series which diverges enter ”DIV” (without quotes).
Then the integrand decomposes into the form
ax + b +
(a)
c
d
+
x−3 x+3
(b)
where
a=
b=
c=
d=
Integrating term by term, we obtain that
�
4x3 + 3x2 − 33x − 30
dx =
x2 − 9
+C
(c)
n=1
∞
1
∑ 3n =
n=2
∞
,
3n
∑ 42n+1 =
,
n=0
∞
xe−3x dx
13.
� (1 pt) Evaluate the indefinite integral.
dx
√
=
x2 x2 + 36
,
,
,
.
16. (1 pt) Match each of the following with the correct statement.
A. The series is absolutely convergent.
C. The series converges, but is not absolutely convergent.
D. The series diverges.
+C.
12. (1 pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -1.
6
11n
10n
∑ n=
n=5 11
∞
6n
(e) ∑ n+4 =
n=1 6
∞
10n + 3n
(f) ∑
=
11n
n=1
(d)
11. (1 pt) Evaluate the indefinite integral.
�
8x2 − 19x − 22
dx =
(x − 2)(x2 − 4)
� ∞
∞
∑ 10n =
1.
∞
∑ (−1)
n=1
∞
n
√
n
n+2
(−3)n
7
n=1 n
∞
(n + 1)(82 − 1)n
3. ∑
82n
n=1
2.
+C.
2
∑
∞
sin(2n)
n2
n=1
∞
(−1)n
5. ∑
n=1 3n + 5
4.
Answer:
∑
Note: Give your answer in interval notation
20. (1 pt) Suppose that
∞
9
=
cn xn
∑
(1 − x3 ) n=0
17. (1 pt) For each of the series below select the letter from a
to c that best applies and the letter from d to k that best applies.
A possible answer is af, for example.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
1.
2.
3.
4.
5.
6.
Find the following coefficients of the power series.
c0 =
The series is absolutely convergent.
The series converges, but not absolutely.
The series diverges.
The alternating series test shows the series converges.
The series is a p-series.
The series is a geometric series.
We can decide whether this series converges by comparison with a p series.
We can decide whether this series converges by comparison with a geometric series.
Partial sums of the series telescope.
The terms of the series do not have limit zero.
None of the above reasons applies to the convergence
or divergence of the series.
∞
2 + sin(n)
∑ √n
n=1
∞
cos2 (nπ)
∑ nπ
n=1
∞
(2n + 3)!
∑ (n!)2
n=1
∞
1
∑ n log(4 + n)
n=2
∞
1
∑ n√n
n=1
∞
cos(nπ)
∑ nπ
n=1
c1 =
c2 =
c3 =
c4 =
Find the radius of convergence R of the power series.
R=
21. (1 pt) The Taylor series of function f (x) = ln(x) at a = 4
is given by:
f (x) =
∞
∑ cn (x − 4)n
n=0
Find the following coefficients:
c0 =
c1 =
c2 =
18. (1 pt) Find all the values of x such that the given series
would converge.
∞
xn
∑ (2)n (√n + 5)
n=1
c3 =
The series is convergent
from x =
, left end included (enter Y or N):
to x =
, right end included (enter Y or N):
Determine the interval of convergence:
c4 =
Note: Give your answer in interval notation
� �
cos 4x2 − 1
22. (1 pt) Let f (x) =
. Evaluate the 6th derivax2
tive of f at x = 0.
f (6) (0) =
Hint: Build a Maclaurin series for f (x) from the series for
cos(x).
19. (1 pt) Find all the values of x such that the given series
would converge.
∞
xn
∑ ln(n + 7)
n=1
3
1.
2.
3.
4.
5.
23. (1 pt) The fourth degree Taylor polynomial for f (x) = x3
centered at a = 2 is T4 (x) = c0 + c1 (x − 2) + c2 (x − 2)2 + c3 (x −
2)3 + c4 (x − 2)4 .
x = 6 cos(t) + cos(4.5t); y = 6 sin(t) − sin(4.5t)
x = sin(t)(3 − 2 sin(t)); y = cos(t)(3 − 2 sin(t))
x = sin(t); y = cos(t) − 2 cos(2t)
x = t 3 − 5t + 2; y = 3 − t 2
x = cos(5t); y = sin(3t)
Find the coefficients of this Taylor polynomial.
c0 =
c1 =
A
c2 =
c3 =
B
C
D
25. (1 pt) A curve with polar equation
c4 =
13
8 sin θ + 65 cos θ
represents a line. This line has a Cartesian equation of the form
r=
? The function f (x) = x3 equals its fourth degree Taylor
polynomial T4 (x) centered at a = 2. Hint: Graph both of them.
If it looks like they are equal, then do the algebra.
y = mx + b ,where m and b are constants. Give the formula
for y in terms of x. For example, if the line had equation
y = 2x + 3 then the answer would be 2 ∗ x + 3 .
24. (1 pt) Assume t is defined for all time. Enter the letter
of the graph below which corresponds to the curve traced by
the parametric equations. Think about the range of x and y, and
whether there is periodicity and or symmetry.
c
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