BC 3
Fall 2014 Semester Review
Name:
The materials in this packet are designed to help guide and focus your preparation for the final. It is not
intended as a complete review. Please look over these problems, plus all of our quizzes and handouts
(lots and lots of these). Many of them you have already done but that does not mean you should not try
them again. Remember that even material that made perfect sense earlier in the semester still needs
review now. Pace yourself while studying during the coming week or so. Do some problems, read and
think through others. Also, be sure to spend time thinking about the conceptual meaning behind the
problems and not just the mechanics of getting through each problem.
Exam:
Wednesday, 12/17, 2:30 pm, in the Academic Pit. Be sure to bring your textbooks!
Topics:
Problems from the book:
Improper Integrals
 Definition
 Evaluating
 p-integrals (two types)
 Tests for Convergence
ch 7
P. 378–9: 7, 9, 11–16, 19, 23, 24, 26, 33, 35, 40, 41
P. 383–4: 1–9, 11–13, 17, 28
Sequences and Series
 Convergence
 Partial sums, geometric series,
telescoping series
 Properties of series
 p-series
 Tests for Convergence/Divergence (lots)
 absolute/conditional convergence
 radius/interval of convergence
 Power, Maclaurin, Taylor series
 Calculating error in using series
(alternating series, Lagrange)
Vectors
 Position, velocity (speed), acceleration
 Modeling trajectories
Polar Curves
 Curve sketching
 Slope of tangent line to polar curve
 Polar Area
Differential Equations
 IVPs, Slope fields
 Euler's Method
 Separation of variables
 Exponential growth/decay, Newton's Law
of Cooling, Logistic growth
ch 8
P. 413–5: some from 1–8, some from 10–15, 20–22,
some from 23–38
ch 9
P. 467–9: some from 7–31, some from 35–57
P. 474–6: some from 1–23, 27, 29, 32, 36
P. 480–1: some from 10–33, 34, 38, 43, 44
P. 488–9: some from 8–43, some from 54–77
P. 496–7: some from 5–21, 22, some from 24–39
ch 10
P. 512–3: some from 1–16, 17, 21–24, 25, 27, 28
P. 517–8: some from 1–24, 28, 30, 32, some from
34–44
P. 523–4: some from 1–21, 22, 23, 25–28, 36
P. 529, some from 1–8, 9, 10, 11, 14, 15,
ch 11
P. 553–4: some from 1–8, 11, some from 13–23
P. 558–60: 4–7, 11–14
P. 563–4: some from 1–9
P. 568–9:, some from 1–27, 28, some from 29–47
P. 576–9: 1, 2, 3, some from 8–26
P. 584–7: 3, 6, 10, 12, 13, 17, 19, 23, 28
P. 594–7: some from 1–12, 14, 16, 20, 22, 23, 24
And don't forget all of the problems on the
handouts!
BC 3 Semester Review.1
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More goodies to do – to complement the problems in the book, not replace them:
(1)
Evaluate the following improper integrals.

x
(a)
dx
2
0
1 x

(2)
2

 
an   
e

(a)
n
2

am  1  
 m
(b)
5
(b)
k 1
(a)

k 1
(c)
 4n
n 1
sin  k 
k

7
(b)
3
 2 m
m


ln  4  3k 
n2
2
1
m 0
(c)
m
  n  1!1
n!
n 1
Write out power series for the following functions
sin  3x 
3
(a) f  x   cos x
(b) g  x  
(c)
x
h  x   xe
x2
(d)
k  x 
x
1  x4
Find the function and the point at which each is evaluated for the given power series.
(a)
2
1    
1   2  2  
2
2
3!
1 2
2
4
0.2   0.2 

1

 
(b)
4!
1 3
Find the interval of convergence for each of the following series:

(a)

k 1
(9)
ak 
ln 1  k 2

 
(8)
x 1
dx
x x
Determine whether each series converges absolutely, converges conditionally, or diverges.

(7)

1
m

3k 1
k 1
(6)

(d)
Determine whether each series converges or diverges. If the series converges, find its limit.

(5)
3
1
dx
x2
Determine whether each sequence converges or not. If the sequence converges, find its limit.
(a)
(4)
10
Determine whether each integral converges or diverges. Justify.
 sin x  2
 2 dx

dx
(a)
(b)
(c)
dx
2
x
3 x  ln x 
1
1 e 1
x3  1

(3)

(b)
If f  x  
 x  4 k



n 0

(b)
3k
k 0
 2 x k
k!
n
 x  2 n

(c)
n
n 1
 3x 
  , find the interval of convergence for f , f , and
 2 
2
 3n
 f  x  dx .
arctan  2 x 
.
x 0
x
(10)
Use Maclaurin series to evaluate lim
(11)
Use series to estimate the value of

0.2
3
xe x dx with error less than 0.00001.
0
BC 3 Semester Review.2
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
(12)
Suppose the power series
 a   x  2
k
k
converges if x = –7 and diverges if x = 9. Which of
k 0
the following statements must be true, which may be true, and which cannot be true? Justify.
(a) Power series converges if x = –11.
(b) Power series converges if x = 5.
(c) Power series diverges if x = –13.
(d) Power series diverges if x = –10.

(13)
Suppose

  x n1 .
n  2n
If 0 < x < 0.3, find the upper bound for the error if 5 terms are used?
If 0 < x < 1, how many terms are necessary if the upper bound for the error is to be less
than 0.001?
If x > 0 and 4 terms are used, how large may x be if the upper bound for the error is to be
less than 0.05?
n 1
(a)
(b)
(c)
(14)
If terms through n = 10 from the Maclaurin series for ex are used to approximate e2, find an
upper bound for the error (don’t use the e key!)
(15)
What values of x can be used if cos  2x  is to be approximated using the first five non-zero
terms of its Maclaurin series if the error is to be less than .005?
(16)
To approximate e with error less than 0.005, how many terms are necessary (again, NO e key!)?
(17)
For x  0.5 , if sin  3x  is approximated using the first four non-zero terms of its Maclaurin
series, how large will the error be?
(18)
Find all points  r ,  , with 0 ≤  < 2 and using exact values, where the tangent line to the
graph of r  2  2sin   is:
(a)
horizontal
(b)
vertical
(19)
Set up the integral or integrals (but DO NOT evaluate) to find the area shared by the graphs of
r  5  4cos   and r  2  2cos   .
(20)
Set up an integral to find the area inside the big loop but outside the small loop of the graph of
r  3  6sin   . Then, approximate this on your calculator.
(21)
Find the equation (in rectangular coordinates and with exact values) of the tangent line to the
curve r  3sin  2  at the tip of the leaf of this graph in the fourth quadrant.
(22)
r
If the position of a particle is given by the vector r  t   2 cos  t  , tan  2t  , find the speed of the
particle at t =
(23)

.
6
r
r
r
r
If a (t )  sin  t  , 2 , v  0   2, 0 and r  0   0, 4 , find r  t  .
BC 3 Semester Review.3
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(24)
From a rooftop 14 m above the ground, a ball is shot out with an angle of elevation of 25 and an
initial speed of 20 m/sec.
r
(a)
Find the position vector r  t  at any time t.
(b)
(c)
(d)
(e)
(25)
(26)
(27)
(28)
(29)
(30)
Find the time at which the ball hits the ground.
Find the horizontal distance traveled by the ball before it hit the ground.
Find the speed of the ball at the time it hits the ground.
Find the maximum height of the ball.
Solve each differential equation.
1
(a) e2 x  y 
y 1 y2
(b)
y  e x  y
Solve the IVP: x  y  y  y 2  1 and y  2   1 .
Consider the differential equation y  y  6  y   5 .
(a) Use technology to sketch a slope field for this differential equation and to sketch solution
curves for each of the following initial conditions. Then discuss the behavior of each
solution curve as t  :
1
(i)
(ii) y  0   4
(iii) y  0  
y  0  7
2
(b)
What is significant about y = 1 for this differential equation?
(c)
Solve the differential equation. (Note: You'll need to do some algebra first!)
Consider the IVP y  y  3 with y  0  2 .
(a) Use Euler’s method with 4 steps (“by hand”) to approximate y(1).
(b) Use Euler method on your calculator to approximate y(1) with 20 steps.
(c) Find the exact solution for y and use it to determine the exact value of y(1).
(d) Are the approximations in parts (a) and (b) too high or too low? Why?
A population P (in thousands) grows so that P  0.0075P  30  P  .
(a) What is the maximum sustainable population under these conditions?
(b) What is the population when the population is growing at its greatest rate?.
(c) If the population is found to be 5000 at t = 0, when will the population reach 8000?
A cool drink is removed from an icebox and is placed in a room where the temperature is a
constant 70F. The temperature of the drink was 35F when it was removed from the
refrigerator and it was 50F 30 minutes later.
(a) What will the temperature of the drink be 75 minutes after it was removed from the icebox?
(b) How long after being removed from the icebox will the drink reach a temperature of 65?
All done. All that’s left is the final exam – and that won’t be nearly as long as the review.
Now, remember to focus on the basics (don’t get too hung up on weird stuff), relax, and stay calm.
Believe it or not, you do know this material, and as Dr. Porzio likes to say, “YOU WILL DO WELL!”
BC 3 Semester Review.4
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