Math 1B
Spring 2016
Final Review
The final exam will be on Wednesday, May 25, 10:30 a.m. – 12:30 p.m. It will cover everything we
have done this semester (chapters 6, 7, 8.1-8.3, 10.1-10.2, 10.4, 11). Remember that you will have 2
hours to complete the exam, so please pace yourself and be aware of the time. You may use one side of
one 8½ in. by 11 in. page of notes. Prepare well and good luck!
1.
Evaluate each integral. If the integral is improper, determine whether it is convergent or
divergent.
3
5
∫ tan x sec x dx
5 x − 12
dx
b) ∫
x + 4x
c) ∫ x€ ln x dx
1
dx
d) ∫
a)
3
e
2
3
1
€
€
€
€
€
x 2 x 2 −1
e) €∫ sin 3 x cos 2 x dx
1
f) ∫
dx
2
x (ln x ) + 4
€
2
1
g) ∫
2 dx
0
( x −1)
1
dx
h) ∫
x 4x + 1
3
1
dx
i) ∫
2 3− x
j) ∫ tan−1 (2x ) dx
(
k)
€ 2.
€
∫e
x
)
1− e 2x dx
Let R be the region enclosed by the curves y = x 2 and y = x 3 .
a) Set up the integral necessary to find the volume of the solid obtained by revolving R about the
x-axis. You do not need to integrate, but identify the type of object made by your typical slice
€
(disk, washer, or shell).
b) Set up the integral necessary to find the volume of the solid obtained by revolving R about the
y-axis. You do not need to integrate, but identify the type of object made by your typical slice
(disk, washer, or shell).
3.
Let R be the region in the first quadrant under the curve f (x) =
x = 1.
a) Find the area of R.
€
€
1
and to the right of the line
x +1
2
b) The region R is revolved about the x-axis. Find the volume of the solid that is generated.
Identify the type of object made by your typical slice (disk, washer, shell).
4.
The part of the curve y = 3 x from (1,1) to (8,2) is revolved about the y-axis. Find the surface
area that is generated.
5.
The region in
bounded by y = e−x and x = 2 is revolved about the y-axis. Find
€ the first quadrant
€
€
the volume of the solid that is generated. Identify the type of object made by your typical slice
(disk, washer, shell).
6.
€
Find the work required to roll up a€40 ft chain ¼ of the way with a 20 lb bucket attached to the
end (the chain weighs 2 lb/ft).
7.
Find the length of the curve y 2 = x 3 from (0,0) to (1,1) .
8.
Find the volume of the solid obtained when the region enclosed by y = x and y = x is revolved
about the line x = €
2 . Identify the€type of€object made by your typical slice (disk, washer, shell).
9.
A tank is 7 feet long, 4 feet high, and has trapezoidal sides that are 6 feet across at the top and 10
feet across at the bottom. The tank is filled with water to a level that is one foot below the top of
the tank. Find the work done in pumping all of the water over the top of the tank. (Recall, the
density of water is 62.5 pounds per cubic foot).
2
10. A milk truck carries milk with density 64.6 lb/ft3 in a horizontal cylindrical tank with diameter 6
ft. Find the force exerted by the milk on one end of the tank when the tank is full.
11. Determine whether the sequence converges or diverges. If it converges, find the limit. Justify
your answers.
∞
' - 1 *$
a) &cos+ (#
% , n )"n =1
b) an = ln( 3n −1) − ln( n + 1)
∞
12. Given the series
€
∑n
n=1
2
10
+ 4n + 3
a) Use the technique of partial fractions to write the general term an of this series as a sum or
difference of two terms.
b) Find a formula for the nth partial sum sn of this series.
c) Evaluate lim sn to determine the sum of this convergent series.
n→∞
13. Determine whether each series is convergent or divergent. If it is convergent, find its sum.
∞
2 2n +1
5 n−1
n= 0
∞
(2n −1)(2n + 1)
b) ∑
16n 2
n=1
a)
€
€
∑ (−1)
n
∞
c)
∑3
2n
n=1
∞
d)
101−n
$ πn + 1 '
∑ sin&% 4n − 3)(
n=1
€
14. Use Taylor/Maclaurin series to find the sum of each series.
€
∞
a)
∑ (−1)
n= 0
∞
€
€
15.
€
n
2n
3n n!
2n +1
(π )
b) ∑ (−1) 2n +1
3 (2n + 1)!
n= 0
2n
∞
n (π )
c) ∑ (−1) n
9 (2n )!
n= 0
n
Determine whether each series is convergent or divergent. Be specific with your answers when
appropriate (absolute vs. conditional convergence). State which test(s) you use.
∞
a)
1
∑ n ln n
2
( )
(2n)!
b) ∑ n
2
n= 0 3 ( n!)
n
∞
(ln n )
c) ∑
n
n=1 (1+ n )
n
∞
−1)
(
d)
n= 2
∞
€
€
∑
n=1
€
€
5
n4
e)
n2 + 9
∑
3
2
n =1 n − n + 1
f)
4 + sin 2 n
∑
n6
n =1
∞
∞
∞
16. Find the radius of convergence and interval of convergence for the series.
7( x − 2)
∑ 3n n
n=1
17. Find the first five nonzero terms of the Taylor series for f ( x) = ln x about a = 1 .
€
n
18. Find the length of the curve given by the parametric equations:
𝑥 = 𝑡!
𝑦 = 𝑡!
0≤𝑡≤ 5
19. Given the parametric equations:
𝑥 = 4 sin! 𝑡
𝑦 = 2 cos 𝑡 − 1
a) Sketch the curve from 𝑡 = 0to 𝑡 = 𝜋. Indicate with an arrow the direction in which the curve
is traced as t increases.
b) Find the equation of the line tangent to this curve when it first crosses the x-axis.
c) Find the area in the first quadrant enclosed by this curve.
20. Sketch the curve and find the area that it encloses:
𝑟 = 1 − sin 𝜃
21. Find the area of the region that lies inside the first curve and outside the second curve:
𝑟 = 3 cos 𝜃
𝑟 = 1 + cos 𝜃
Answers to Final Review
1.
€
€
€
i) Improper integral; Convergent, 2
1
j) x tan−1 (2x ) − ln(1+ 4 x 2 ) + C
4
1 −1 x 1 x
k) sin (e ) + e 1− e 2x + C
2
2
€
€
1
1
sec 7 x − sec 5 x + C
7
5
& x # 3
b) 2 ln$
!+ +C
% x+4" x
3 4 1
e +
c)
16
16
2
x −1
+C
d)
x
1
1
e) cos5 x − cos 3 x + C
5
3
1 −1# ln x &
f) tan %
(+C
$ 2 '
2
g) Improper integral; Divergent
h) ln 4 x + 1 − 1 − ln 4 x + 1 + 1 + C
a)
2.
€
€
€
a)
b)
1
∫ π (x
∫ π (y
0
1
0
− x 6 ) dx
4
2
3
)
− y dy
3.
€
4.
€
€
π
4
π2 π
b)
−
8 4
a)
π
145 145 −10 10
27
(
5.
π (1− e−4 )
6.
900 ft-lb
7.
13 2 − 8
27
8.
8π
15
9.
28,875 ft-lb
€
)
3
€
€ 10. 5480 lb
11. a) Converges to 1.
b) Converges to ln3.
5
5
−
n +1 n + 3
5 5
5
5
b) sn = + −
−
2 3 n+2 n+3
25
c)
6
12. a) an =
50
9
b) Diverges (Test for Divergence)
c) Geometric; Converges to 90
d) Diverges (Test for Divergence)
13. a) Geometric; Converges to
14. a) e −
2
3
3
2
1
c) cos( π3 ) =
2
b) sin( π3 ) =
€15.
€
a) Convergent (Integral Test)
b) Diverges (Ratio Test)
c) Absolutely convergent (Root Test)
d) Conditionally convergent (Alternating Series Test, p-series)
1
e) Convergent (Limit Comparison Test w/ 2 )
n
f) Convergent (Comparison Test)
16. R=3, I = [−1,5)
€
1
2
17.
€
( x −1) − ( x −1)
18.
335
27
€
2
+
1
1
1
3
4
5
( x −1) − ( x −1) + ( x −1)
3
4
5
19. a) The curve lies along the parabola 𝑥 = 4 − 𝑦 − 1
!
!
b) 𝑦 = − ! 𝑥 + !
5
c)
3
20.
3π
2
21. 𝜋
!