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MATH 1B FINAL (PRACTICE 1)
PROFESSOR PAULIN
DO NOT TURN OVER UNTIL INSTRUCTED TO DO SO.
CALCULATORS ARE NOT PERMITTED
THIS EXAM WILL BE ELECTRONICALLY SCANNED. MAKESURE YOU WRITE ALL SOLUTIONS IN THE SPACES
PROVIDED. YOU MAY WRITE SOLUTIONS ON THE BLANKPAGE AT THE BACK BUT BE SURE TO CLEARLY LABEL
THEM!
tan(x) dx = ln | sec(x)|+ C
!sec(x) dx = ln | sec(x) + tan(x)|+ C
cos2(x) =1 + cos(2x)
2sin2(x) =
1− cos(2x)
2
|EMidn | ≤ K(b− a)3
24n2|ESn | ≤
K(b− a)5
180n4
ex = 1 + x+ x2
2+ x3
6+ · · · =
"∞n=0
xn
n!
sin x = x− x3
3!+ x5
5!− x7
7!+ x9
9!− · · · =
"∞n=0(−1)n x2n+1
(2n+1)!
cos x = 1− x2
2!+ x4
4!− x6
6!+ x8
8!− · · · =
"∞n=0(−1)n x2n
(2n)!
arctan x = x− x3
3+ x5
5− x7
7+ x9
9− · · · =
"∞n=0(−1)n x2n+1
2n+1
ln(1 + x) = x− x2
2+ x3
3− x4
4+ x5
5− · · · =
"∞n=1(−1)n−1 xn
n
(1 + x)k = 1 + kx+ k(k−1)2!
x2 + k(k−1)(k−2)3!
x3 + · · · ="∞
n=0
#kn
$xn
limn→∞(n+1n)n = e
Name:
Student ID:
GSI’s name:
Math 1B Final (Practice 1)
This exam consists of 10 questions. Answer the questions in thespaces provided.
1. Compute the following integrals:
(a) (10 points) !x2 ln(x3)dx
Solution:
PLEASE TURN OVER
Math 1B Final (Practice 1), Page 2 of 12
(b) (15 points) ! √x2 − 9
x4dx
Solution:
PLEASE TURN OVER
81
8T
81 81 81
8T
81
Math 1B Final (Practice 1), Page 3 of 12
2. (25 points) Calculate the area of the surface of revolution (around the x-axis) of thecurve
y = (x− 1)3,
between x = 1 to x = 2.
Solution:
PLEASE TURN OVER
Math 1B Final (Practice 1), Page 4 of 12
3. (25 points) Determine if the following series are convergent or divergent. You do notneed to show your working.
(a)∞%
n=2
(1
n+ 1− 1
n)
Solution:
(b)∞%
n=1
(−1)n cos(1/n3)
Solution:
(c)∞%
n=1
n+ 1
n4 − 10
Solution:
(d)∞%
n=1
5n
3n + 4n
Solution:
(e)∞%
n=1
nn
n!
Solution:
PLEASE TURN OVER
Convergent
Divergent
Convergent
Divergent
Divergent
Math 1B Final (Practice 1), Page 5 of 12
4. (a) (20 points) Using the integral test, determine whether
∞%
n=1
1√ne
√n
is convergent or divergent. Be sure to check all the hypotheses of the integral test.
Solution:
PLEASE TURN OVER
I1 Cx TE e
y Fck O ou I
y 7 continuous on I
31 Tx et 0 and increasing on Ct 7 Cx decreasingon Ct N
et u Tx DIE doc Zira du
Jrater du 2J et du 2e CZ
C
I erd Eis Ira fine It
Ie convergent
T
jMeru convergent
Integral Test
Math 1B Final (Practice 1), Page 6 of 12
(b) (5 points) Using this, or otherwise, determine if the infinite series
∞%
n=1
1
nen
is convergent or divergent.
Solution:
PLEASE TURN OVER
0Cute E Fm all u Irue s
T 1
Hence Teru convergent L Ten convergent
T n In I
Comparison Test
Math 1B Final (Practice 1), Page 7 of 12
5. Let
f(x) =(x2 + 1)
ex2 .
(a) (20 points) Calculate the Maclaurin series of f(x). Be sure to include a generalterm.
Solution:
(b) (5 points) Using this, or otherwise, determine the value of f (2n)(0).
Solution:
PLEASE TURN OVER
e I txt t I
d I x2t C 1Z I n l
G22 et
tXl 24 u I Zn
t x21 f I tD h l lXlt l y u n l
I t C 1 C l 2 ue n t
n
for all u in C a ay t
1 2 1
Maclaurin series ofex 2
7 Col Can Zu Zn
Math 1B Final (Practice 1), Page 8 of 12
6. (25 points) Consider the differential equation y′ = 9− y2. Graph the solutions with thefollowing initial conditions: y(0) = −2, y(0) = 1 and y(0) = −4. Be sure to label eachsolution carefully.
Solution:
PLEASE TURN OVER
i
y co 4
Math 1B Final (Practice 1), Page 9 of 12
7. (25 points) Find an equation for the orthogonal trajectory to the family of curves
y =1
k − x+ 1 (k any constant)
which contains the point (1, 2).
Solution:
PLEASE TURN OVER
Math 1B Final (Practice 1), Page 10 of 12
8. (25 points) Solve the following initial value problem:
t3y′ + 2y = 1 y(1) = e+ 1/2.
Solution:
PLEASE TURN OVER
Math 1B Final (Practice 1), Page 11 of 12
9. (25 points) Find a general solution to the differential equation:
y′′ + y = x sin(x).
Solution:
PLEASE TURN OVER
y t y o r't 1 0 r Ii
Yu C cos cxltczsia.cn c general homogeneoussolution
Jp CA ox Apt cos x Box 1 B x Siu Gil
Yp Ao 12A c cos x Ao x 1 A x2 sin x
Bo 2B x sin x Dox 1131 2 cosec
Tp 2A six Hot 2A a Siu Gil CAO 12A c Siu x
Ao x A x2 ans x 2B Sin X Do 1213 c caste
Bo x t B x2 sin xBo 1 2B sc Cns LM
y p t yp GA 2130 t 413 a lzesiucal
ZAO 213 L A X Siu x
2 It 12130 0 A 4 general413 o Bo T solutionZAO 1213 O B O
4A I Ao O
y C cos cxltczsiu.cn t 4 x2cascx1 2csiuG4
Math 1B Final (Practice 1), Page 12 of 12
10. (25 points) Find a power series solution to the following initial value problem:
y′′ + xy′ + 2y = 0, y(0) = 0, y′(0) = 1
Solution:
END OF EXAM
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