Page 1 Calculus II : Project 1 /30faculty.madisoncollege.edu/alehnen/calculus2/projcalc2s17.pdf ·...

Page 1 Calculus II : Project 1 /30Name Due 1/24/17________________________ "Þ Ð7.5 points)Compute the following derivatives for the given functions. The Wolfram Alpha website can be used to check answers. The link at can be accessed from the course websitehttp://www.wolframalpha.com http://faculty.matcmadison.edu/alehnen/Calculus2/Calculus_2_Home_Spring_17.html .

a) forf f ÐBÑ œ B B ± ÐBÑ œ 1 1 È* ± $2 w ________________________

b) for 0 f f ÐBÑ œ B Á ÐBÑ œsinBB

w ________________________

c) forf f ÐBÑ œ BÑ B ± Ÿ ÐBÑ œ cos cos 1 Š 1Ð$ ± $‹ w ________________________

d) f f ÐBÑ œ B ÐBÑ œ e # $B w ________________________

e) for or f f ÐBÑ œ B ÐBÑ œ ln 1 % B ! B #Š ‹#B

w ________________________

Page 2 2 2. (7.5 points)Evaluate the following integrals. The evaluatesWolfram Research On Line Integrator at http://integrals.wolfram.com/index.jspindefinite integrals symbolically, but it does not supply the arbitrary constant, hence its results may differ by an algebraicrearrangement from any other correct answer. The WinPlot program or a graphing calculator can evaluate definite integrals

numerically. To use Winplot to evaluate the definite integral, use the 2-dim Window. Enter the integrand, i.e., ,' ,

+0ÐBÑ 0ÐBÑdB

under the1. Explicit y = f(x) Equa format. From the View menu choose Vew/Set corners to set less than and larger than Theleft right+ ,. down up and values depend on the extremes of on . From the View menu, Grid may be chosen to set convenient tick0ÐBÑ Ò+ß ,Ómarks and tick labels. To numerically evaluate the definite integral choose Measurement/Integration from the One menu. Set thelower limit to and the upper limit to . Choose the desired number of subintervals (i.e., the value for the numerical quadrat+ , 8 ure),select the approximation method to be used (left endpoint, midpoint, right endpoint, trapezoidal, parabolic (Simpson's rule) andrandom) and then press the definite button to see the numerical result. Answers to indefinite integrals can also be checked this way byseeing if the calculated value of the associated definite integral with chosen upper and lower limits agrees with the numerical definiteintegral (see ).http://www.execpc.com/~aplehnen/calc2pr2.htm

a) ' Š% cos sin B $ B B œ d ‹%

________________________

b) ' È)> $ dt œ ________________________

c) ' !

"-

ln 1 ee 1Ð B

B

Ñ d B œ ________________________

d) ' sinÐ B Ñ

B

ÈÈ d B œ ________________________

e) ' #

" ÐB$Ñ

B

2d B œ ________________________

Page 3 3

3. 7.5 points Evaluate the following limits.Ð Ñ

a) lim B Ä !

e 1sin( )#B

B% œ ________________________

b) lim tan tanh( ) B Ä ∞

-1Š B œ‹ ________________________

c) lim B Ä !

B œB ________________________

d) lim B Ä "

È È2 1BB

BÐ BÑ

4 3 B È$ œ ________________________

e) lim tan( B Ä "

ÐB "Ñ# 1# BÑ œ ________________________

Page 4 4

4. 7.5 points)Ð

Given C f œ ÐBÑ œ BeB2

a) Compute f wÐBÑ œ ________________________

b) Compute f wwÐBÑ œ ________________________

c) Locate the roots of f ÐBÑ .

d) Find the location of all the maixma and minima of Indicate which points are maxima (and whether they are absolute or o f ÐBÑ . nlylocal ) and which are minima (absolute or local ).

e) What is the area under the curve from to the location of the first positive maximum?B œ ! B

f) lim e E Ä ∞

' E

!B œB2

.B ________________________

g) Find the position of any inflection points of B ÐBÑ .f

h) Sketch below:f ÐBÑ C œ BeB2

Page 5 5

Self-Assessment: (2 bonus points)a) Describe three strengths in your perfomance on this project. Include why each is a strength.

b) What are three things that could be improved about your performance on this project. Explain specifically how you will makethese improvements.

c) Identify two things about this project which are still unclear to you.

d) Identify two insights that you have acquired in doing this project.

Page 6 Calculus II : Project 2 /30Name Due 2/09/17________________________ 1. Evaluate the following integrals (18 points)Þ

a) ' ŠB #‹2e B d B œ ________________________

b) ' e sin #B Ð%BÑ B œd ________________________

c) ' 4

"sec "Ð BÑÈ d B œ ________________________

d) ' 1

!cos sin Ð$BÑ Ð B# Ñ d B œ ________________________

Page 7 2

e) ' / Ð/B Bsec Ñ B œd ________________________

f) ' 1%

!cos $Ð#BÑ B œd ________________________

g) ' csc $Ð"BÑ B œd ________________________

h) ' sec 6Ð ÑB# d B œ ________________________

Page 8 3

i) ' 1#

!tan 'Ð ÑB# d B œ ________________________

j) ' tan ) sin ( ) # #Ð# #) ) d ) œ ________________________

k) ' #1

!> " Ð>ÑÈ cos d > œ ________________________

l) ' "

!È$' *B# .B œ ________________________

Page 9 4

m) ' $

"

"È"%%B%%$'B# .B œ ________________________

n) ' "È%B #%B# .B œ ________________________

o) ' BB B"$#' .B œ ________________________

p) ' " B "BÈ d B œ ________________________

Page 10 5

q) ' BB B&#' .B œ ________________________

r) ' B B B"'B B

$ #

$' (

% .B œ ________________________

2. (2 points)a) Evaluate both of the definite integrals below. Give both an analytical (i.e., an 'exact' formula) and a numerical answer.

Analytical Answer Numerical Answer (Use a Calculator)

' e

"ln ÐB Ñ B œB d ________________________ ________________________œ

' 1

!B B B œ2sin d ________________________ ________________________œ

b) Use a numerical quadrature (integration) method (Simpson's Rule would be a good choice) to check your answers.

Numerical Quadrature Answer

' e

"ln ÐB Ñ B œB d ________________________

' 1

!B B B œ2sin d ________________________

c) How well do the numerical quadrature procedures approximate the definite integrals?

Page 11 6

3. (3 points)

a) Evaluate the definite integral ' "

!

.B "B# œ ________________________

In order to make the numerical quadrature below more painless, you should use a computer to perform the calculations.

b) Estimate the number of intervals, needed in order to approximate this integral to within an absolute error of using 8ß 0.0001 theTrapezoidal Rule.

c) Compute the Trapezoidal Rule approximation to this integral using the value of estimated in part a). (8 Please don't do this 'byhand'! )

d) From your answer to part b), determine the actual error of the trapezoidal approximation. Is the error smaller than in0.000"absolute value?

e) Now approximate this same integral using the Trapezoidal Midpoint and Simpson's Rule. For all approximations start with 4ßsubdivisions of the interval of integration (i.e., and keep doubling the number of subdivisions until either the approximate8 œ %Ñintegral is within one one millionth of its last estimate or subdivisions have been used.128

8 Trapezoidal Rule Midpoint Rule Simpson's Rule48163264128

f) Which approximation converges the fastest. Explain your answer.

4. (2 points)The Dominated Convergence Theorem for improper integrals states that if for all , then 0 Ÿ g f aÐBÑ Ÿ ÐBÑ B

' ' ' '∞ ∞ ∞ ∞

a a a ag d f d f d g dÐBÑ B ÐBÑ B ÐBÑ B ÐBÑ B converges if converges ; diverges if diverges

a) What can you say about if diverges? Support your answer with an example.' '∞ ∞

a ag d f dÐBÑ B ÐBÑ B

b) What can you say about if converges? Support your answer with an example.' '∞ ∞

a af d g dÐBÑ B ÐBÑ B

Page 12 7

5. (5 points)Give an argument that shows whether the following improper integrals either converge or diverge.

a) ' ∞

1"B(È .B

b) ' ∞

!

"È/ BB .B

c) ' ∞

!

"È/ BB .B

d) ' ∞

!

sin (# BÑB# .B

e) '∞∞

"cosh( sin( )BÑ B .B

f) ' ∞

"’ “1# ÐBÑ .Btan"

g) [ : Integrate by parts.]' ∞

"

sin(BÑB .B Hint

h) ' ∞

0

sin(BÑB .B

i) ' ∞

∞

sin(BÑB .B

j) [ : ]' ∞

"

l BÑlsin(B .B l ÐBÑl ÐBÑ sin sin Hint #

Page 13 8





Page 14

Instructions for Group WorkGroup Projects 1 through 5 all are to be done in groups. The following guidelines should be adhered to in forming your group, doingthe work and writing up the project.

Group Requirements:

Each group must consist of at least two individuals but no more than four individuals. You are free to form your own groups, but ifyou can't find a partner see me and I'll assign you to a group. Some class time will be devoted to group work, but much of it will haveto be done outside of class. It is up to the group to decide any internal division of labor, eg., who is responsible for what parts of theproblem, who will be the 'algebra expert' , who will check the work, who will write up what parts of the report. It is possible one grouphas one individual write the entire problem, while in another group everyone writes up different parts. It is in your own best interest toinsist that you understand the solution of the whole problem. You are free to use any written resources or computing technology insolving the problem.

Report Requirements:

Each group must hand in one solution for a given project which should include the following :

1. The names of all group participants. the report writers feel an his/her assigned task, you are If individual did not perform free todelete that person's name from the report. I will arbitrate all appeals on such disagreements and reserve the right to give either awritten or oral exam to decide the issue.

2. The conclusions of any questions stated neatly in complete sentences which are both concise and complete in expressing theanswers.

3. The mathematical work to each problem attached in a way which is both neat and clear. Solutions should be presented in the reportin the same order that the associated problems appear in the project.

Grading:

1. Each person in the group will receive the same point total out of the point value of the group portion of the project. Thus, it is theresponsibility of everyone in the group to review the answers to all of the questions.

2. Grades will be based on the mathematical correctness both of the results and the methods used to arrive at them. Thus, a rightanswer arrived at by accident using faulty mathematics will not count for much. Points will be deducted for incomplete, illegible,sloppy or incomprehensible answers.

Page 15 Calculus II : Group Project 1 ( 8 points ) Due 2/17/17 Name _______________________ Name _______________________

Name _______________________ Name _______________________

Choose either the pair of problems 1 and 2 or all of Problem 3 . You may earn up to 8 bonus points by doing all three problems.

1. (4 points)Introduction to the (Gamma) Function>

By its definition the Gamma function is : for e 0>Ð5Ñ œ > .> 5 ' ∞5" >

! .

Evaluate the following :

a) b) > >Ð"Ñ œ Ð#Ñ œ __________________ __________________

c) d) > >Ð$Ñ œ Ð%Ñ œ __________________ __________________

e) f) > >Ð&Ñ œ Ð'Ñ œ __________________ __________________

g) Develop a recurrence formula between and { : Integrate by parts > >Ð5 "Ñ Ð5Ñ . Hint ×

h) Combining the information in part a) with that in part g) , determine the formula for for any positive integer >Ð8Ñ 8 .


Name _______________________ Name _______________________

2. (4 points)Some Results for Improper Integrals

a) Give an example of a function such that diverges, but converges.0ÐBÑ ' '∞ ∞

0 0| f | d f dÐBÑ B ÐBÑ B c d#

b) Suppose that converges, must also converge? If so, provide some justification, if not, provide a' '∞ ∞

0 0| f | d f dÐBÑ B ÐBÑ Bc d#

counter example.

c) Suppose that converges, must also converge? If so, provide some justification, if not, provide a' '∞ ∞

"| f | d dÐBÑ B B

1

f ÐBÑB

counter example.

d) Suppose that converges and What is the value of ? Justify your answer.' ∞

"| f | dÐBÑ B lim

B Ä ∞0ÐBÑ œ P PÞ


Name _______________________ Name _______________________

3. (8 points)Introduction to the Laplace Transform

For positive the Laplace Transform of a function is defined as Determine the Laplace= 0ÐBÑ P 0ÐBÑß = œ / 0Ð>ÑÒ Ó ' ∞

0=> d> Þ

Transforms for the six functions below .

a) 0ÐBÑ œ " P 0ÐBÑß = œ Ò Ó ________________________

b) 0ÐBÑ œ B P 0ÐBÑß = œ Ò Ó ________________________

c) 0ÐBÑ œ B P 0ÐBÑß = œ8 Ò Ó ________________________

d) for0ÐBÑ œ / P 0ÐBÑß = œ+B Ò Ó ________________________ ( + = Ñ

e) 0ÐBÑ œ Ð+BÑ P 0ÐBÑß = œsin Ò Ó ________________________

f) 0ÐBÑ œ Ð+BÑ P 0ÐBÑß = œcos Ò Ó ________________________

g) Is the Laplace Transform a linear operator, i.e., does ? Explain.P +0ÐBÑ ,1ÐBÑß = œ +P 0ÐBÑß = ,P 1ÐBÑß =Ò Ó Ò Ó Ò Ó

h) Without perfoming an integral determine P Ð+BÑß = œÒ Ósinh ________________________

Page 18 2 i) Assume is differentiable to at least second order and has the properties that for any positive CÐBÑ = / CÐBÑ œ !

B Ä ∞ lim =B

limB Ä ∞

/ C ÐBÑ œ !=B w

Obtain an expression for in terms of and P C ÐBÑß = CÐ!Ñ P CÐBÑß =Ò Ó Ò Ó Þw

P C ÐBÑß = œÒ Ów ________________________

j) Under the assumptions of part i) obtain an expression for in terms of andP C ÐBÑß = CÐ!Ñ ß C Ð!Ñ P CÐBÑß =Ò Ó Ò Ó Þww w

P C ÐBÑß = œÒ Óww ________________________

k) Consider the following second order differential equation subject to the stated initial conditions :

with C $C #C œ "! ÐBÑ ß CÐ!Ñ œ "ß C Ð!Ñ œ %ww w wcos .

By taking the Laplace transform of both sides of this equation solve for .P CÐBÑß =Ò Ó

P CÐBÑß = œÒ Ó ________________________

l) Using the technique of partial fractions decompose into a sum of partial fractions.P CÐBÑß =Ò Ó

P CÐBÑß = œÒ Ó ________________________

m) Finally, using the results of the previous page determine the function which has this Laplace transform. (You have thusCÐBÑsolved the differential equation!)

CÐBÑ œ ________________________

Page 19 Calculus II : Project 3 /30Name Due 2/27/17________________________ 1. (4 points)Indicate which of the following sequences converge, and for convergent sequences evaluate the limit. Öa8× a) a8 œ " Ð Ñ8

b) a8 œÐ Ñ1 8

8

c) a8 œ Š #1È ‹8

d) a8 œ 8"Þ"

)

8

e) a8 œ 8 Ð Ñln 1 B8

f) a 8 œ tan tan 1 1Ð $8 Ñ 8ÑÐ'

g) a8 œ 8 BÈ8 # 8

h) a8 œ " Ð B8

8Ñ

i) a8 œ 8x e8

j) a8 œÐ Ñ1 8 Ð8 Ñ

Ð8"Ñ1 x

Ð%8#&Ñ x#

Page 20 2

2. (2 points)Some of the following statements are always true (i.e., they are theorems), others are not. For each statement, indicate whether or notit is a theorem. If it is not a theorem, provide a counter example. The sequence is understood to consist only of real numb{ }a8 ers.

a) If and then lima a8 8 ! ß8 Ä ∞

œ P P !ß Þ

b) If is bounded , then converges{ } { }a a8 8 Þ

c) If is not bounded, then diverges{ } { }a a8 8 .

d) If is bounded and , then converges{ } { }a a a a8 8 8" 8 Þ

e) If is not bounded and , then .{ } lima a a a8 8 8" 8

8 Ä ∞œ ∞

f) If is decreasing, then converges{ } { }a a8 8 Þ

g) If is decreasing and , then converges{ } { }a a a8 8 8 ! Þ

h) If is neither increasing nor decreasing, then diverges{ } { }a a8 8 Þ

i) If and , then a a8 8 ! œ P lim8 Ä ∞

Ð "Ñ P œ !8 Þ

Page 21 3

3. (1 point) Landen's transformation of elliptic integrals uses the sequence defined by the following recurrence relation:

5 œ8"# 55 "

È 8

8

a If , show that this sequence is increasing and bounded above by .Ñ ! "5 "!

b) If ,! œ8 Ä ∞

5 "! 8 klim __________________

c) If If , explain whether or not the limit of this sequence must still exist.5 "!

4. (5 points) Evaluate the following sums.

a) Š∞

8œ"$ 1 18 8 % ‹ œ __________________

b) Š∞

5œ"

1 1È È%5 %5% ‹ œ __________________

c) Š∞

4œ"

" "4" 4 tan tan " "Ð Ñ Ð Ñ ‹ œ __________________

d) ∞

8œ!

#!& Ð 8 ÑÐ 8(Ñ2 2 œ __________________

e) ∞

4œ#

' % j jj j

2

2 2# Ð Ñ1 œ __________________

Page 22 4

5. (10 points) For each of the following indicate (with a valid justification) whether the series is absolutely convergent, conditionallyconvergent, or divergent.

a) ∞

8œ"

Ð sin 812 Ñ

8&È

b) for any real number∞

8œ!" sin(8BÑ

82 B Þ

c) ∞

8œ"

Ð Ñ 1 8

8 % #È8 '8 )

d) ∞

8œ#

8Ñ ln(8#

e) ∞

8œ"

Ð"Ñ Ð# xx

8 8ÑÐ#8 Ñ1

f) ∞

8œ"

Ð Ñ 2 8

# 88 $

g) ∞

8œ!

Ð" !Ñ 00 #8 &!8

8xÈ

Page 23 5

h) Š∞

8œ"8 8 1 1 sin ‹

i) ∞

8œ" Ð Ñ1 81 É% #

)8

j) where and ∞

8œ# a a a8 # 8" œ œ 1

ln 2 1 ln 1ln

2a8 Ð8 8Ñ

Ð8 Ñ Ð8 Ñ

6. (1 point) Consider the sequence + œ " 8 Š ‹"8

8.

a) Analytically evaluate lim

8 Ä ∞œ+8 __________________

b) Using a calculator or a computer program calculate to at least 7 decimal places. a a&! &! œ __________________

c) How well does approximate the actual limit?a&!

Page 24 6

d) Now consider the sequence , analytically evaluate , œ ,8 8"#!8 #$ #8""#!8 "$ #8"

#

# Š ‹8 lim

8 Ä ∞œ __________________

e) Using a calculator or a computer program calculate to at least 7 decimal places. , ,"" "" œ __________________

f) How well does approximate the actual limit?,""

g) Now, consider the infinite series Give a convincing (to me, that is) argument that this sum converges.∞

8œ!

1 8x .

h) Using your calculator, compute the sums to at least 7 decimal places for through WR œ R œ ! R œ ""R

8œ!

"8x Þ

R R R R

! $ ' *

" % ( "!

# & ) ""

The Partial Sum The Partial Sum The Partial Sum The Partial Sum W W W WR R R R

i) How do these sums compare with the numerical values of and and ? lim 1 8 Ä ∞

Ð 18 Ñ + ,8 &! ""

7. (1 point) .The harmonious Euler Constant #

Consider the infinite series 1 where and # œ œ œ∞

8œ1+1 1

Ð Ñ8 8 B

8

8Ba a a8 2 2

18" 8 ' d

a) Does this series converge? Justify your answer.

b) Write the partial sum in terms of the sum of the first terms in the harmonic series and the ln function.S28 8

c) Evaluate lim8 Ä ∞

1 + + ... +ln

1 1 1 1# $ % &

Ð

18

8"Ñ

Page 25 7

8. ( : 4 points) .Bonus Problem Building a Stairway to HeavenConsider identical uniform solid blocks each of length and weight . The blocks are stacked on top of each other to lean out8 P 71over the edge of a table. The side of length of each block is perpendicular to the table's edge. The bottom block leans out a P distanceP P#8 #Ð8"Ñ from the table's edge. The second block from the bottom leans out a distance from the edge of the bottom block. This

pattern is repeated for each block, i.e., block , which is on top of block , leans out a distance from the edge o 4 4 " P#Ò8Ð4"ÑÓ f

block . Hence, the top block leans out a distance from the edge of block . This is illustrated below for 4 " 8 " 8 œ &P# Þ

a) What is the average moment (i.e., where is the center of gravity) of the blocks as measured from the table's edge?8

b) Let stand for the horizontal distance of the leading edge (the edge away from the table) of the top block from the table'sH8 edge.Determine an expression for .H8

c) Determine the smallest value of for which .8 H P8

d) lim 8 Ä ∞

œD8 __________________

Page 26 8

9. (2 points) The Riemann zeta function is defined by the following infinite series:

'ÐBÑ œ∞

"4œ

"4B

a) For what values of does this series converge?B

From the theory of Fourier Series, (or see Problem 6 of Group Project 3) one can establish that

...'Ð#Ñ œ œ "Þ'%%*$%!'')%)##'%$'%(#S œ∞

"4œ

"42

2 œ Þ

16

Furthermore by an integral approximation using trapezoids one can bound for a decreasing convex function as followsß

∞

"4œ0Ð4Ñ 0ÐBÑ :

R ∞ R

" " "4œ 4œ 4œ

0ÐRÑ#0Ð4Ñ 0ÐBÑ.B 0Ð4Ñ 0Ð4Ñ 0ÐBÑ.B ' '∞ ∞

R R

b Using a calculator or a computer evaluate and from this calculate Lower and Upper Bounds to Ñ Þ &!

"4œ

"42 S

&!

"4œ

"42 œ ____________ ___________ ____________ Lower Bound to Upper Bound to S S œ œ

c How well do these bounds approximate the actual value of ?Ñ S

d) Estimate the number of terms required to evaluate to within 'Ð$Ñ "! Þ '

e) Estimate to within 'Ð$Ñ "! Þ '

10. (4 points) Some of the following statements are always true (i.e., they are theorems), others are not. For each statement, indicatewhether or not it is a theorem. If it is not a theorem, provide a counter example. The sequence is understood to consist on{ }a8 ly of

real numbers and means the infinite series of the with the lower limit unstated but understood as a definite number .∞

8a8 a8

a) If converges then ∞

8a a8 8ß Þlim

8 Ä ∞œ !

Page 27 9

b) If then converges lim8 Ä ∞

œ !a a8 8ß Þ

∞

8

c) For any real Bß B Á "ß"

"B

∞

8œ!œ B Þ8

d) If diverges and diverges, then diverges.∞ ∞ ∞

8 8 8a b a8 8 8 8Ð , Ñ

e) If diverges , then both and diverge.∞ ∞ ∞

8 8 8Ða a b 8 8 8 8 , Ñ

f) If diverges , then at least one of and diverges.∞ ∞ ∞

8 8 8Ða a b 8 8 8 8 , Ñ

g) If the partial sums of are bounded , then converges.R

8 8

∞

a a 8 8

h) If the partial sums of are bounded, then converges.R

8 8

∞

|a | a 8 8

i) If but is not an eventually decreasing sequence, then diverges.a a a 8 8 8 ! ß { }∞

8

j) If converges, then converges.∞ ∞

8 8a a8 8Ð "Ñ8

k) If converges, then converges.∞ ∞

8 8la a8 8l Ð "Ñ8

Page 28 10

l) If both and converges, then converges.∞ ∞ ∞

8 8 8a a a8 8 8Ð "Ñ8 l l

m) If converges, then converges.∞ ∞

8 88|a |8a8

n) If converges, then converges.∞ ∞

8 8

l8a8a8l

o) If and , then converges.a a a8 8 8 ! lim8 Ä ∞

œ ! Ð "Ñ∞

8

8

p) If converges, then converges.∞ ∞

8 8a a8 8

#Ð Ñ

q) If converges, then converges.∞ ∞

8 8l Ða a8 8

#l Ñ





Page 29 Calculus II : Project 4 /30Name Due 3/06/17________________________ 1. (5 points) For each power series below : (1) Determine the center, radius, and interval of absolute convergence. (2) If the radius of convergence is finite, determine if the series converges at the endpoints.

a) ∞

8œ"

sinŠ812 ‹ B

8*"

8

b) ∞

8œÐ

1 Ð8"Ñx B

8"Ñx

8

c) On this problem you need not determine the convergence at the end points. You may earn two bonus points∞

8œ!

Ð x 8 B ÑÐ#8 Ñx

8 #

1

for a correct analysis of convergence at the end points.

d) ∞

7œ" Èm ) Bm

e) ∞

8œ(

Ð$ B'Ñ8

8$

$

2. 1 point)Ð

Given the following functions, generate the infinite Taylor series about the point . State enough terms so that the pattern is a apparent.Note: our text uses a rather "non-standard" notation in calling the point about which the function is expanded as " ". Most othe- rauthors designate this point as " " and use " " to stand for the point where the derivative in the remainder is be evaluated. In + - theseproblems we will follow the "standard" notation and let be the point about which the function is expanded.+

a) ; 1f a ÐBÑ œ $B B 'B " œ 3 2 ""

Page 30 2

b) ; cos f a ÐBÑ œ #B Ñ œÐ !

c) ; f a ÐBÑ œ œ "eB

3. (2 points)Using Winplot generate and hand in a plot on the interval of the cosine and its five Taylor polynomials (partial sums) [ , 2 ]! 1 ofdegree 2 , 4 , 6 , 8 and 12. In Winplot is expressed as fact( ).8

a) On the interval , how well is the cosine approximated by its 12'th degree Taylor's polynomial?[ , 2 ]! 1

b) As you added more terms did the approximating polynomial 'break away' sooner or later from the cosine?

c) Display the remainder, for the 'th degree Taylor polynomial to about R n .8ÐBÑ B B œ , cos !

d) Determine the limit of as R8ÐBÑ 8 Ä ∞ .

Now use the same plotting program to generate and hand in a second plot on the interval of and its five Taylor [0 , 2] ln 1Ð ÑB polynomials (partial sums) of degree 2 , 4 , 6 , 8 and 12.

e) All the Taylor polynomial approximations to appear 'to fail' at about the same point in the graph. Would using evenlnÐB "Ñhigher degree Taylor polynomials improve the approximation at or beyond this point? Explain your answer.

Page 31 3

4. (6 points)Express each function below as a power series in State enough terms so that the pattern is apparent.B .

a) cosÐB Ñ œ$

b) sinhÐ ÑB# œ

c) B%eB œ

d) d dB Š B%eB‹ œ

e) cos sinB B B œ

f) ' B

!cosÐ > œÈ Ñ dt

5. (8 points)Using your knowledge of power series evaluate the following infinite series.

a) ∞

8œ!

Ñ ( 1 ( )8 #8"$#1

Ð#8"Ñx œ __________________

b) ∞

4œ! ( )$

#1 #4"

Ð#4"Ñx œ __________________

Page 32 4

c) ∞

5œ"

5# 5 œ __________________

d) ∞

5œ"

"5# 5 œ __________________

e) ∞

5œ"

"5x#

5 œ __________________

f) B B $ B B B B B B#x $x %x &x 'x (x

& ( * "" "$ "&

ÞÞÞ œ __________________

g) B B B B B B B B# % ' ) "! "# "% "'

# % ' ) "! "# "% "'

ÞÞÞ œ __________________

h) 1 4 16 64 256 1024

! ! ! ! !B B B B B# % ' ) "!

# % ' ) "!

ÞÞÞ œ __________________

'. (4 points)For each problem below evaluate the limit by the use of an appropriate series.

a) lim B Ä !

1 cosh BB2 œ __________________

b) lim B Ä !

sinhsin

Ð+BÑÐ,BÑ œ __________________

Page 33 5

c) lim ln 1 B Ä ∞

B Ð Ñ œ &B __________________

[ : make an expansion in ] Hint 1B

d) lim B Ä ∞

")B (( ÐB B B *Ñ œŠ ‹È 2 __________________

(. (2 points)

a) The indefinite integral can not be expressed in terms of a finite number of elementary functions. Using Simpson's rul' eB#.B e

or any other numerical quadrature scheme on a computer or calculator calculate the following definite integral until successiveapproximations differ by no more than 1 ‚ "!' Þ

'!Þ&!

eB#.B œ __________________

Page 34 6

b) By making use of the convergent Maclaurin series of the exponential function , express the above definite integral as an infiniteseries, then evaluate the first six terms of this series.

'!Þ&!

e B#.B œ __________________

c) Estimate the maximum error incurred by truncating the series after only six terms. __________________

d) Which method , numerical quadrature or series expansion, seems preferable for this problem?

8. ( 2 points)Bonus Problem:

Suppose you need to know the numerical value of the definite integral to the nearest . You could use a'∞!

B"

ee

B

B .B ‚ "!1 8

numerical quadrature scheme, but this could be time consuming given the required accuracy. An alternative is to find the analytic(exact) answer, but there is no elementary function whose derivative is the integrand. Nevertheless, you need to calculate both theanalytic and numerical answer to this integral. [ Change variables, then use a power series expansion.]Hint:

a) Analytic Answer œ __________________

b) Numerical Answer œ __________________

Page 35 7

9. (2 points)A piece of 1a) What is the Maclaurin series for ?tan1B

b) For what real values of does this series converge to ?B tan1B

c) Let and . Show that . [ Use the addition formula for .]α " 1 α " α " tan tan 4 tanœ œ œ Ð Ñ Ð Ñ 1 1Ð Ñ Ð Ñ" "

# $ Hint:

d) Using a calculator, estimate and by summing the arctangent series up through and including the 11'th order term.α "

The Partial Sum Estimate of The Partial Sum Estimate of α " ____________ ____________œ œ

e) Estimate based on your estimates of and . The Partial Sum Estimate of 1 α " 1 ____________œ

f) Based on the fact that the terms in the arctangent expansion alternate in sign, estimate the maximum error in part e) .

g) Let and . Show that # $ 1 # $ tan tan 4 4œ œ œ Ð Ñ 1 1Ð Ñ Ð Ñ" "& #$*

h) If you were to approximate and by summing the arctangent expression up through and including the 11'th order term,# $ would you obtain a less or more accurate approximation to than in part e) ? Explain.1

Page 36 8

10. ( : 2 points)Bonus Problem

If and both converge, then the product of the two absolutely convergent series can be rearranged as the "Ca∞ ∞

8œ! 8œ!l , l - 8 8l l uchy

Product" shown below.

∞ ∞ ∞ 8

8œ! 7œ! 8œ! 7œ! , ,8 7 87 7- œ -

Since the tangent is an odd function, the Maclaurin series for consists of only odd powered terms.tanÐBÑ

tanÐBÑ œ + B∞

8œ0 8

#8"

However, the standard Maclaurin series formula for computing the is extremely cumbersome owing to the complexity of theÖ+ ×8derivatives involved. As an alternative, use the identity that In terms of power series this is expressed sin cos tan . ÐBÑ œ ÐBÑ † ÐBÑ as

∞ ∞

8œ! 4œ!

Ð"Ñ Ð"ÑÐ#8"Ñx Ð#4Ñx

8 4

B œ B + B#8" #4 #8"∞

8œ08

If you then use the Cauchy product of the two power series on the right hand side of this formula, you can obtain a recurence formulafor the . In any case, determine the values of the following coefficients.+8

_____________________+ œ!

_____________________+ œ"

_____________________+ œ#

_____________________+ œ$






Name _______________________ Name _______________________

Choose any two of the following four problems. All problems are worth 4 points. You may earn additional bonus points by doingextra problems.1 ..The Deft FlyTwo bicycles speed towards each other each at a speed and separated initially by a distance . An extremely agile fly initially@ Pperched on one of the handle bars begins flying (hence the species name) back and forth between the two bikes at a speed .- - @, The fly leaves for its first flight at the precisely the same instant that the bikes begin their ominous trek towards annihilation. The fly,defying (de-flying?) all known laws of physics, makes instant turnarounds with no changes in speed. The two bikes eventually crash,injuring the bikers and terminating forever the illustrious career of said fly.

a) Let stand for the distance the fly traveled on its first inter-bike flight. Let stand for the time of this flight.H >" "

What is the relationship between and ?H" >"

Solve for in terms of and >" - @ P, , .

Solve for in terms of and D c v L1 , , .

b) Let stand for the distance the fly traveled on its th inter-bike flight. Let stand for the time of this flight.H >8 88w

What is the relationship between and ?H >8 8

Obtain the recursion formula for in terms of and .> >8 8"- @, ,

Obtain the recursion formula for in terms of and .H H8 8"- @, ,

c) Obtain the explicit formula for in terms of and .> P8 - @, ,

d) Obtain the explicit formula for in terms of and H P8 , , - @ Þ

e) Write the total time of the fly's flight as an infinite series and then sum the series to get a formula in and - @, , PÞ

f) Write the total distance of the fly's flight as an infinite series and then sum the series to get a formula in , and .- @, P

g) How long would it take two bikes initially a distance apart and each moving towards the other at a speed to meet?P @

h) How far would a fly moving at a speed travel in this amount of time?c


Name _______________________ Name _______________________

2. The Koch Snowflake.Construct a sequence of regular polygons is an equilateral triangle of side 1. is obtained from byX X X X X X X X! " # $ % ! " !ß ß ß ß ÞÞÞ taking each side (or edge of length dividing into three equal segments and then replacing the middle third of each side by aP! Ñß n'outward facing' equilateral triangle of side is obtained by the same construction applied to , and in general is"

$P X! # " 8"Þ X X obtained by the same construction applied to Let be the length of each side of be the number of sides of beX P X X8 8 8 8 8 8Þ ß W ß T the perimeter of and be the area of X E X8 8 8ß .

a) Find explicit formulas for each of the following :

W œ œ œ8 8 8___________ ___________ ___________P T

b) lim 8 Ä ∞

T8 œ __________________

c) E! œ __________________

d) Calculate the increase in area in going from X X8 8" 8" 8to Þ E E œ __________________

e) Find an explicit formula for E E œ8 8. __________________

f) lim

8 Ä ∞E8 œ __________________

g) Is there anything peculiar about results to questions b) and f) ? Explain.

h) The Hausdorff-Besicovitch dimension of a figure can be defined as H H œ 8 Ä ∞lim ln(

ln(W Ñ

Ñ8

8P ÞCalculate for the Koch Snowflake.H

H œ __________________


Name _______________________ Name _______________________ 3. Rearrangements of the Alternating Harmonic Series.Riemann showed that by suitably changing the order of summation in a conditionally convergent series one can change the answer to

any desired real number. Probably, the most famous conditionally convergent series is the alternating harmonic series , ∞

"4œ

Ð"Ñ4

4"

which summed in this order converges to ln( .#Ñ œ !Þ'*$"%(")!&&**%&$!*%"(#$#ÞÞÞ

a) Consider the following rearrangement of the alternating harmonic series gotten by subtracting the next two reciprocal evennumbers after each reciprocal odd. What is the value of ?E

E œ œ"" " " " " " " " " " "# % $ ' ) & "! "# ( "% "' ÞÞÞ ‘∞

"8œ

" " "#8" %8# %8

[ First, can be calculated without rearrangement asHint: E

E œ Ð Ñ Ð Ñ Ð Ñ Ð Ñ ÞÞÞ" " " " " " " " " " " "# % $ ' ) & "! "# ( "% "'

Seond, factor out ]"#

___________E œ

b) Consider a second rearrangement of the alternating harmonic series. What is the value of ?F

F œ œ"" " " " " " " " " " "$ # & ( % * "" ' "$ "& ) ÞÞÞ ‘∞

"8œ

" " "%8$ %8" #8

[ : ConsiderHint W œ "

" " " " " " " " " " "# $ % & ' ( ) * "! "" "# ÞÞÞ

]" " " " " " "

# # % ' ) "! "#W œ ! ! ! ! ! ! ÞÞÞ

F œ ___________

c) Find your own rearrangement of the alternating harmonic series which converges to a negative number. Explain yourrearrangement and provide compelling evidence that the sum is indeed negative.

d) Consider the infinte series , which in this stated order converges to Can you find a rear∞

"4œ

Ð"Ñ4

4"

# 0.8224670... . rangement

of this series which converges to a negative number? Explain.


Name _______________________ Name _______________________

4. the Riemann zeta functionProof of Euler's Identity that 'Ð#Ñ œ 1#

' ('(BÑ )Cosider a polynomial of degree for which zero is not a root. 8 T ÐBÑ œ + B + B + B ÞÞÞ + B +

8 8#8"

8 8" 8#" ! ß

with + Á Þ! 0a) Based on your knowledge of Algebra, justify the identity that T ÐBÑ œ T Ð!ÑÐ" ÑÐ" ÑÐ" ÑÞÞÞÐ" Ñ

B B B B< < < <" # $ 8

where are the (in general complex) roots of , , , ..., < < < < 8 T ÐBÑ" 2 3 n ß .

b) Write each of the following polynomials as the product of linear factors of the form of part a .8

T ÐBÑ œ %B #% œ __________________

T ÐBÑ œ #B B "&# œ __________________

T ÐBÑ œ ÐB $ÑÐB "'Ñ# œ __________________

c) Consider What are the real roots of ?0ÐBÑ œ 0ÐBÑB Á !

" B œ ! sinÐBÑ

B

d) Assuming that can be treated as a polynomial write as a product in the form of part a) .0ÐBÑ 0ÐBÑ

e) Expanding the infinite product of part d one gets that where is given by an infinte series.0ÐBÑ œ " -B SÐB Ñ -# % ß

The series for . - is œ __________________

f) What is the power series about for ?B œ ! 0ÐBÑ

g) Equating the coefficients between parts e) and f) leads to what infinite series identity?B#

Page 41 Calculus II : Project 5 /30Name Due 3/30/17________________________ 1. (9 points)For each first order differential equation below: Indicatei. If it is a separable differential equation. ii. If it is linear differential equation.Then solve the differential equation subject to the stated initial conditions.

a) with .0.B '

"œ B $B 0Ð "Ñ œ Þ#

b) with .C.B œ BC C ÐBÑ CÐ Ñ œ # Þ# #cos 1

c) with .0.B œ #B0ÐBÑ 0Ð!Ñ œ ' Þ

d) with for

for

.C

.B œ 1ÐBÑ œ CÐ!Ñ œ " Þ$ B Ÿ #

B " B #Ö

Page 42 2

e) with when .BC #C œ B C œ B œ "w # , ""&

f) with .0.B œ %/ $0ÐBÑ 0Ð!Ñ œ # Þ B2

2. (1 point)

a) Solve subject to .C.B #BC œ B CÐ!Ñ œ + Þ

b) Generate the direction field graph for this equation with a window from to in both the and directions. On this sa #Þ& #Þ& B C megraph plot the five functions for and CÐBÑ + œ !Þ& ß + œ ! ß + œ !Þ& ß + œ " + œ "Þ& ÞIn WinPlot use the 2-dim Window and from the Equa menu select Differential dy/dt. In the differential equation menuu, set equx' al

to and equal to Select the "vectors" check box. Change the window using the command sequence View , Set corners . Ini1 y' .C.B . tial

values can be specified through the One/dy/dt trajectory menu. Just set equal to zero (for this example) and equal to ignx y (+ ore thesetting for , then press the "draw" button with "both" checked. Repeat this procedure for each initial condition. Choosing the t) IVPs adifferent color from the slope field makes the display easier to interpret.

c) Explain what happens at + œ !Þ& Þ

Page 43 3

3. (1 point)Find and classify (as stable or unstable) the equilibrium solutions of the following differential equations. Use the Differential Equaoption of Winplot to investigate stability.

a) .B.>.C.>

œ !Þ!"B !Þ!#C

œ !Þ!#BC !Þ!$C

b) [ Let and write this second order ODE as a system of first order ODE's.]. B .B .B.> .> .>

#

# ' "$B œ ! C œ Hint:

4. (1 point) Linear Circuits:a) A resistor and an inductor are hooked up in series to a power supply which establishes aV Ð&Þ! Ñ P Ð!Þ"&! Z Ð&Þ! ÑH H) volts steady state current of 1.0 amperes . The power supply is then removed and the current decays in a manner described by :Z

V Ð Ñ

V3 P œ ! Þ !Þ#&.3.> How long after the power supply is removed is the current down to amperes? Note : ""

HH œ " second .

For arbitrary values of and , how long after the power supply is removed is the current down to ?V ß P Z Z%V

b) A capacitor is charged to a potential difference . The power supply is removed and the capacitor isG Ð#Þ! Z Ð"!Þ!.F) volts)allowed to discharge through a resistor M . The charge on the capacitor is given by and decays in a mannerV U U œ GZÐ"Þ& ÑH

described by : . How long after the power supply is removed is the charge on the capacitor down to itsV œ ! !Þ#&.U U.> G

starting value? For arbitrary values of and (the power supply voltage), how long after theNote : 1F second† " œ "H Þ V ß G Z power supply is removed is the charge down to ?GZ

%

Page 44 4

5. (2 points) Concentration DilutionÞa) A tank of volume has a chlorine concentration of . It is desired to reduce this to by pumping in water with a chloriZ - -! 0 neconcentration of . To prevent over flowing water is pumped out at the same rate. If the water is pumped in at a rate- Ð- - - ÑM ! M0

of gallons per minute and if the water is so well mixed that at any time the chlorine concentration is uniform through out the< tank ,how long will it take to reach the desired concentration?

b) How long would it take if 'pure' water were used instead of the water with a chlorine concentration of ?-M

6. (3 points) .Population Growtha) The most naive (and most frightening) model for population growth is the exponential growth model. This assumes that the rate ofgrowth is directly proportional to the current population size. In symbols this is expressed as , where is the numbe.R

.> œ 5R R rof individuals in the population at time and is a constant. Roughly, is the number of new 'offspring' produced by each> 5 5

individual in a relevant unit of time. Solve , subject to the initial condition that .R.> œ 5R RÐ!Ñ œ R Þ!

b) The exponential model is obviously unrealistic for large times in that it leads to unbounded population growth. There is only somuch matter in the universe! At some point limited food and living space limit the population size of any species. One way to modelthis behavior mathematically is to modify the rate equation as follows : , where is the limiting population.R R

.> Pœ 5RÐ" Ñ P

size. Initially increases exponentially as before; however, as approaches the rate of growth slows to zero. Hence, is R R P R œ P ahorizontal asymptote of the solution. This rate equation is sometimes called the logistic equation.For the logistic equation model what value of makes the rate of growth a maximum?R

c) What is this maximum rate of growth? How does this compare to the rate of growth of the exponential model for the samepopulation size?

d) Solve the logistic equation, subject to the initial condition that RÐ!Ñ œ R Þ!

Page 45 5

e) Evaluate lim > Ä ∞

RÐ>Ñ œ ________________________

f) For and according to the exponential model how long would it take the populationR œ $!! ß P œ &!!ß !!! 5 œ !Þ"&! year , "

to double to ? To increase to ?'!! "!!ß !!!

g) For and according to the logistic model how long would it take the population toR œ $!! ß P œ &!!ß !!! 5 œ !Þ"&! year , "

double to ? To increase to ?'!! "!!ß !!!

h) For and make a careful graph of the solutions of both the exponential and logisticR œ $!! ß P œ &!!ß !!! 5 œ !Þ"&! year ,"

models. A graphing calculator or computer program would be helpful here.

Page 46 6

7. (2 points)Perform the following operations on the given complex numbers and express the result in standard rectangular form (+ ,3ÑÞ

a) & #& È œ __________________

b) Ð) (3Ñ Ð $ "' ÑÈ œ __________________

c) Ð"! '3ÑÐ& * ÑÈ œ __________________

d) (*3#& "'È È œ __________________

8. (1 point)Cosider the equation . Using Euler's identity three times and equating the real and imaginary parts of both sid/ / œ /3 3 3 α " α "† ( ) esresults in what trigonometric identity?

9. (1 point)For real expressB

a) in terms of sin sinhÐ3BÑ ÐBÑÞ

b) in terms of cos coshÐ3BÑ ÐBÑÞ

c) in terms of sinh sinÐ3BÑ ÐBÑÞ

d) in terms of cosh cosÐ3BÑ ÐBÑÞ

Page 47 7

10. (2 points)For each of the following complex numbers i) Express the result in standard rectangular form ii) Express the result in standard exponential form ( </3) Ñ

a) Rectangular ExponentialÐ# 3ÑÐ$ 3Ñ œ œ __________________ __________________( ) ( )

b) Rectangular ExponentialÐ$/ ÑÐ#/ Ñ3 3% #1 1

œ œ __________________ __________________( ) )Ð

Page 48 8

11. (7 points)Solve the following initial value problems. Each problem is worth 1 point.a) with C C 'C œ ! CÐ!Ñ œ &ß C Ð!Ñ œ &ww w w .

b) with . 0 .0 .0.> .> .>

#

# % "$0 œ %! Ð$>Ñ 0Ð!Ñ œ # ß œ "cos l!

c) with C %C %C œ $#/ CÐ!Ñ œ %ß C Ð!Ñ œ $ww w w#B .

d) with C $C %C œ "!/ CÐ!Ñ œ #ß C Ð!Ñ œ 'ww w % wB .

Page 49 9

e) with C )C "'C œ #/ ÐBÑ CÐ!Ñ œ $ß C Ð!Ñ œ "#ww w % wBsin .

f) with C %C œ "'B Ð#BÑ CÐ!Ñ œ #ß C Ð!Ñ œ %ww wsin .

g) with C #C C œ %)B / CÐ!Ñ œ #ß C Ð!Ñ œ $ww w # wB .






Name _______________________ Name _______________________

Choose any two of the following five problems. You may earn up to 12 bonus points by doing all problems.1. (4 points)a) Money flows continuously into an account at a fixed rate of dollars per year. If the account is intially set up with a5balance of and if the account earns interest at a fixed annual rate of , how much money is in the account after years?T < >

b) How much of the money in the account after years is earned interest?>

c) An account is opened with an initial investment of and per year continuously flows into the account. If the$ $$!!! "#!!account earns continuously compounded interest at an annual rate of , how much money is in the account after years?&Þ# &%


Name _______________________ Name _______________________

2. . (4 points)The Logistic Model with HarvestingLet represent the population of a species at time , the rate of proportional growth, the limiting population in the absRÐ>Ñ > 5 P ence ofany harvesting, and the rate of harvesting, i.e., the number of individuals removed per unit of time. This leads to the ODEV

.R R.> Pœ 5RÐ" Ñ V (1)

subject to the initial condition that the population when harvesting began. Since by definition if everRÐ Ñ œ R RÐ>Ñ ! RÐ>Ñ0 , ! ß

equals zero the species has become extinct and the value of for which this happens is the extinction time.>

a) In the limit as for what values of is extinction inevitable? For in this domain determine a formula for theP Ä ∞ R R! ! extinction time.

b) For finite there is a critical value of called such that if extinction becomes inevitable for any initial populatiP V V V V - - on.Determine a formula for in terms of andV 5 PÞ-

c) For and finite determine the equilibrium solutions of equation and classify each solution as stable or unstable.V V P- (1)

d) For and finite for what values of is extinction inevitable? For in this domain determine a formula for theV V P R R- ! ! extinction time.


Name _______________________ Name _______________________

3. Consider the differential equation , with the initial condition dfdB œ ÐBÑ Ðc df f# .!Ñ œ +

Express as the power series 0ÐBÑ .∞

8œ! - B8

8

a) What is ? -0

b) What is the power series for ?0 ÐBÑw

c) From the differential equation obtain the recursion formula for in terms of - - - - - -8 8" 8# 8$ !ß ß ß ÞÞÞ ß ß Þ 1Ò ÞHint: Use the Cauchy product ]

d) Solve this recursion relation to obtain an explicit formula for as a function of -n n .

e) Write the explicit power series for 0ÐBÑ .

f) What function is this?

g) Solve the differential equation by separation of variables.

h) What conclusion can you now draw?


Name _______________________ Name _______________________ 4. .Transcient CircuitPVGA capacitor, an inductor, , and a resistor, are hooked up in series. The capacitor is charged to a DC voltage of aG P V Z , , ! nd thenallowed to discharge through the resistor and inductor. The current, , satifies the second order, linear homogeneous ODE :M

with and P V M œ ! ß MÐ!Ñ œ ! M Ð!Ñ œ . M.>

.M "

.> GZP

#

#w ! Þ

If , one has an circuit in which the current varies sinusoidally at the resonance frequency If V œ ! PG œ V Á ! ß=!"PGÈ Þ

V can never be negative) the current decays exponentially in time.a) What is the condition on in terms of the values of and so that the current is underdamped ?V P G

b) What is the current as a function of time when the circuit is underdamped ?

c) What is the current as a function of time when the circuit is critically damped ?

d) What is the current as a function of time when the circuit is overdamped ?

e) If and calculate the value of the resistance for critical damping . _________________P œ %Þ) G œ "Þ# V œ mH F .

f) For volts and each value of in the table below, calculate the missing parameters inP œ %Þ) G œ "Þ# Z œ "!Þ! VmH F ß ß. !

the units requested. Here If 1456.89 sec 1 45689 msec= =œ Þ œ œ œÉ " V ‚"!PG %P " " ‚"! œ

#

# $

$" "1456.89 1456.89

sec sec Þ Þ

V Ð Ñ

!Þ!!

"Þ!!

&Þ!!

"!Þ!

&!Þ!

"!!Þ

H = =(sec ) (msec ) (sec ) (msec ) (amps)" " " "V V#P #P

Z

!

Ê P VG %

#

g) Generate on 1 plot six graphs of current versus time for volts and each value of inP œ $Þ% G œ "Þ# Z œ "!Þ! VmH F ß ß. !

the table above. For all graphs plot the current from to .! "Þ'msec msec


Name _______________________ Name _______________________ 5. Driven CircuitPVGA capacitor, an inductor, , and a resistor, are hooked up in series with an AC power supply. The current, , satifiesG P V M , , thesecond order, linear ODE : where is the AC voltage function Suppose for an angular velocity P V M œ ß Z Z œ Z Ð >Ñ. M

.>.M " .Z.> G .>

#

# Þ !cos = =

and an amplitude fixed by the power supply. Replace by the complex voltage and solve the resulting complex ODE byZ Z Z /! ! 3 >=

assuming a particular complex solution of the form . Solve for in terms of and The particular solutionE/ E P G V ß Z ß3 >= ß ß ! = . for the actual current is then given by .MÐ>Ñ œ E/Real Part of ( 3 >= Ñ

b) What is the amplitude of the driven current ?

c) Relative to the power supply voltage what is the phase shift of the current ?

d) What value of gives the most current for all other circuit constants held fixed ? This is called the resonnance frequency o= f thecircuit.

e) Explain why in this analysis we ignored the homogeneous solution.

Page 55 Calculus II : Project 6 /30Name Due 4/19/17________________________ You can use Winplot to graph both parametric as well as polar plots. Choose the 2-dim Window and in the Equa menu pick 2.Parametric for parametric plots x = f(t), y = g(t) or 4. Polar for polar plots r = f(t) .

1. (3 points) Given the parametric equations : B œ ' >Ñ #cos Ð sin C œ & >Ñ $ Ð a) Sketch the parametric curve for ! Ÿ # > Ÿ 1

b) Compute at the point d dd dC C&B B Š È$ # # $ , ________________________È2

‹ œ

c) Compute at this same point. d dd d

2 2

2 2C CB B œ ________________________

d) What is the eccentricty of this conic section?

Page 56 2

2. (3 points)Given the following parametric equations B œ / >Þ">cos a) Sketch the curve for ! Ÿ > Ÿ ) C œ / >1 Þ">sin

b) Find the equation of the tangent line to the point Ð"ß !Ñ Þ

c) Find the arc length of the curve for .! Ÿ > Ÿ )1

d) Find the arc length of the curve for .! Ÿ > ∞

e) Calculate the area in the first quadrant bounded on the outside by this curve for ! Ÿ > Ÿ 1# .

3. (2 points)Given the following parametric equations B œ + >cos$a) Sketch the curve for and + œ # ∞ > ∞ C œ + > sin$

Page 57 3

b) Find the slope of the curve at any point on the curve.ÐBß CÑ

c) Consider the point what is the equation of the tangent line to the curve at this point?Ð Ñ$+ $) )

+Èß ß

d) Let be any point on this curve in the first quadrant. What is the length of the segment on the tangent line to the curve atT whichTlies between the and axes?B C

4. (2 points)a) A particle moves in an orbit described by the equations À B œ + Ð >Ñcos = sin C œ + Ð >Ñ Þ=

Calculate the arc length, , swept out in a time L T œ #1= .

P œ ________________________

Page 58 4

b) A particle moves in an orbit described by the equations À B œ + Ð >Ñcos = sin C œ , Ð >Ñ Þ=

Calculate by numerical integration (answers to the nearest ten-thousandth) what the ratio of the arc length , , (swept out in a L time

X + . œ#1= ) to is for different values of the ratio You may use any means at your disposal and to your liking to e œ " É ,

+

#

#

perform the numerical integration. Fill in the table below.

e œ "

!Þ!

!Þ"

!Þ#

!Þ$

!Þ%

!Þ&

!Þ'

!Þ(

!Þ)

!Þ*

"Þ!

É ,+ + # +

#

#P P

1

c) Calculate the area, , enclosed by the orbit of part b).E

d) Evaluate the following limits:

lim _______________________

e Ä !E œ

lim _______________________e Ä "

E œ

e) What does mean about the orbit? Does the result for at agree with this observation? Does the result for ate eœ ! œ !P+ E

e œ ! agree with this observation?

f) What does mean about the orbit? Does the result for at agree with this observation? Does the result for at e e eœ " œ " œ "P+ E

agree with this observation?

Page 59 5

5. (2 points)A circle of radius rolls without slipping on the outside of a circle of radius . Let be the angle from the positive axi b a > B s to the linethat connects the centers of the two circles. Consider the specific point on the outer circle (imagine it is painted on the ciT rcle),which is the point of tangency when {i.e., when has coordinates }. 0 0 0 > œ > œ T , a , Ð Ña) Show (or at least try to show) that for any angle, the coordinates of the point are given by the parametric equat > T , Ð Bß C Ñ ions ofthe epicycloid : a b b a b bBÐ>Ñ œ Ð Ñ > CÐ>Ñ œ Ð Ñ >cos cos sin sin> >Š Ša b a b

b b ‹ ‹ ;

b) Using a computer or calculator plot the unit circle and the epicycloid having and Rather than sketching thea b œ œ1 . "%

graph by hand you may instead hand in a computer plot.

Graph of Epicycloid Graph of Hypocycloid

c) For the case just considered in part b), how many full rotations does the rolling circle undergo in making one full revolution aboutthe stationary circle? Justify your answer from the equations and don't be fooled by the graph. Explain why the graph looks this way.

d) If the rolling circle of radius moves without slipping on the inside of a larger circle of radius then the coordinates ob a , f lie onTthe curve generated by the parametric equations of the hypocycloid : a b b a b bBÐ>Ñ œ Ð Ñ > CÐ>Ñ œ Ð Ñ >cos cos sin sin> >Š Ša b a b

b b ‹ ‹ ;

Page 60 6

Using a computer or calculator plot the unit circle and the hypocloid having and Rather than sketching the grapha b œ œ1 . "%

by hand you may instead hand in a computer plot.

e) For the case just considered in part d), how many full rotations does the rolling circle undergo in making one full revolutioninside the stationary circle? Justify your answer from the equations and don't be fooled by the graph. Explain why the graph looksthe way it does.

6. Sketch the following polar curves. (4 points) a) < # œ sin )

b) < œ # $ cos )

Page 61 7

c) < œ &$#sin )

d) < Ð$ Ñ œ sin )

7. (2 points)a) What is the total area enclosed by the figure in problem 6. d) ?

b) What is the area in a single ' petal ' of the figure in problem 6. d) ?

c) Set up and numerically evaluate the integral which gives the length of the full curve in problem 6. d)

Page 62 8

8. (1 point)Find all points of intersection of the curves defined by and< œ % < œ " # cos cos .) )

9. ( 5 points)Bonus Problem Using a computer plotting program generate and attach to your project five polar graphs. Three should be from the list of ' required 'curves, and two from the list of ' optional ' curves. In Winplot the view window should be set as follows :

left down R : right up R : lowt 0 : hight L*pi œ œ œ œ œ œ

Here R is the maximum value of and is the smallest integer that gives a 'complete' curve (i.e., for angles larger than ± < ± , L L1 thecurve just retraces over that which was drawn from to ). and must be determined for the list of required curves, but are! L R L1supplied for the optional ones.

Required Curves Optional Curves R L r sin t/8) r cos(11t) 1 1 œ Ð œ r exp(sin(4t)) r cos(50t)*(1+sin(t)) 2 2 œ œ r 1 2sin(t/2) r 1 2sin(t/4) 3 8œ œ r 2 7cos(sin(t)+sin(121t)) 9 2œ r 1 cos(8t ^9 2 2œ Ñ r sin(t/5)/(2+cos(t/8)) 1 80œ r = 2 cos(50t) 3 2

To graph implicit curves in Winplot choose the 2-dim Window and in the Equa menu pick 3. Implicit. After entering the formula for0ÐBß CÑ 0ÐBß CÑ œ click on to display those coordinates within the viewing window where .ß ok constant

10. (4 points)For each of the following quadratic curves : (1) Identify which conic section or degenerate case it represents. (2) Calculate (if not a parabola) and , .a b c (3) Calculate the eccentricity, e . (4) Calculate the coordinates of all vertices and/or centers. (5) Calculate the coordinates of all foci. (6) Write the equation of the directrix. (7) Sketch the curve, showing asymptotes if it is a hyperbola.

Page 63 9

a) % " CB *C B *! #!& œ2 2 6 0

b) $' *C (#B CB &% $'* œ2 2 0

Page 64 10

c) 0#C )C2 "'B %! œ

d) $'C *B (#C B2 2 &% $'* œ 0

Page 65 11

11. (3 points) Write the Cartesian coordinate equation of a conic section having the given directrix, focus and eccentricity.a) Focus : Directrix : Eccentricity , Ð$ #Ñ œ #B œ "

b) Focus : Directrix : Eccentricity , Ð " $Ñ C œ # œ"#

c) Focus : Directrix : EccentricityÐ% &Ñ B œ , 1 œ #

12. (1 point)a) What curve is generated by the parametric equations for B œ Ð>Ñ sec 1

2 2> 1

C œ Ð>Ñtan

b) Of course, as both and with the same behavior at . Taking this into account, what curve tan sec> > > Ä Ä „∞ , , 12 2

31

do these parametric equations generate for ?0 2Ÿ > 1

Page 66 12

"3. (3 points)a) Find the Cartesian equation of < œ &

$1+ sin )

b) Find the polar equation of a conic section of eccentricity, , focus at the origin and directrix e œ B Þ"# œ #

c) Find the polar equation of a conic section of eccentricity, focus at the origin and directrix e .œ œ " , $# C






Name _______________________ Name _______________________

Choose any one of the following two problems. Each problem is worth 8 points. You may earn up to 8 bonus points by doing bothproblems.1. Polar functions of the form are called limicons (pronounced lee-ma-shanns, !) Using a computer¸< >œ a b cos not lee-mi-konsplotting program plot the following six limicons all on the same graph. Since the curves below have all been normalized by div¸ idingthe numerator by its maximum absolute value, all of them will fit in the same window. Attach your graph with the project. a) r (1 .3cos(t))/1.3œ b) r (1+.5cos(t))/1.5œ c) r (1+.7cos(t))/1.7œ d) r (1+ cos(t))/œ Þ* "Þ* e) r (1+cos(t))/2œ f) r (1+5cos(t))/6œ a) Under what conditions, for and does the limicon, have an inner loop?¸ , , a b a b ! ! œ cos< >

b) For and and , evaluate at at and . a b a b ! ! œ cos < > > œ ! > œ > œ > œ.C.B # #

$ ß ß ß1 11

c) For and and , find the values of where the curve has horizontal tangents. a b a b ! ! œ cos < > >

d) For and and , find the values of where the curve has vertical tangents. a b a b ! ! cos< œ > >

e) For what values of and , is there a ' dimple ' in the curve at ? a b a b ! cos < œ > > œ 1

f) How would the graph of change if cosine were replaced by sine.< >œ a b cos


Name _______________________ Name _______________________

2. Four bugs are placed at the four corners of a square of side . Each bug crawls counterclockwise at the same speed directl + ytowards the next bug to its left. The path of the four bugs is shown below.

a) Taking the center of the square as the origin find the polar equation of the path for the bug that started at the top right corner.[ The line from each bug to the next bug to its left is tangent to the first bug's path.]Hint:

b) What are the polar equations for the paths of the other three bugs?

c) Find the distance travelled by each bug in going from its corner of the square to the center.

Page 69 Calculus II : Project 7 /39Name Due 5/05/17________________________ 1. (2 points) For each vector below, find its length, its direction, and the angles and that± V V

Ä Ä± œ , , s V V

V

Ä

± ±Ä α " # , ,

makes with the positive and axes, respectively. B D , C

a) V 4 5 V VÄ Ä

œ # $ & à ± ± œ œ às 3s s s ____________ ____________ ____________à α œ " œ ____________ # œ ____________

b) V 4 5 V VÄ Ä

œ ‚ Ð Ñ à ± ± œ œ às 3s s s ____________ ____________ ____________à α œ " œ ____________ # œ ____________

c) V 5 V VÄ Ä

œ ‚ à ± ± œ œ às 3 4s s s ____________ ____________ ____________à α œ " œ ____________ # œ ____________

d) VÄ

œ ‚ Ð ‚ Ñ à ± ± œ œ àÄ s 5 5 V Vs s 3 4s s ____________ ____________ ____________à α œ

" œ ____________ # œ ____________

2. (5 points) Perform the following vector operations:a) (% 3 3s s s ss s ' & Ñ Ð$ # ( Ñ œ4 5 4 5 ________________________

b) ( # 3 3s s s ss s $ & Ñ Ð % $ ' Ñ œ4 5 4 5† ________________________

c) ( 3 3s s s ss s Ñ ‚ Ð# # $ Ñ œ# $4 5 4 5 ________________________

Page 70 2

d) ( # #3 3 3s s s s s ss s s # $ Ñ ‚ Ð Ñ ‚ Ð# # Ñ œ4 5 4 5 4 5Ð Ñ ________________________

e) Ð Ñ( # #3 3 3s s s s s ss s s # $ Ñ ‚ Ð Ñ ‚ Ð# # Ñ œ4 5 4 5 4 5 ________________________

3. (1 point)a) Find the area of the parallelogram whose vertices are at the pointsÐ " ! " Ð % " " Ð # # " Ð & $ " , , ) , , , ) , , , ) , , , ) .

b) Find the volume of the parallelepiped whose vertices are at the pointsÐ " ! " Ð % " " Ð # # " Ð & $ " Ð $ " " Ð ' # " Ð % $ " Ð ( % , , ) , , , ) , , , ) , , , ), , , ) , , , ) , , , ) , and , ) " .

4. (5 points) Give the equations for the following geometrical objects :a) A sphere of radius centered at the point % # $ $( , , ) .

b) A parametric equation for the line passing through and .Ð # $ ( " % # Ñ , , ) ( , ,

c) The plane containing , and .( , , , , , , , # " $ Ñ Ð " " % Ñ Ð # $ & Ñ

d) A parametric equation for the line of intersection of the two planes : and B C #B # #D œ # C #D œ % .

e) The angle between these same two planes.

Page 71 3

5. (1 point)Consider the plane , defined by the equation and the point with coordinates .ß U *B "#C #!D œ %! T Ð "% ß #( ß %# Ña) Find a parametric equation for the line through normal to T U Þ

b) Where does this normal line through intersect ?T U

c) What is the distance from to ?T U

6. (1 point) Let be the line with coordinates satisfying the equation , and let be the line with_ `Ð Bß Cß D ÑB" D## $ %

C"œ œ

Ð Bß Cß D Ñ coordinates satisfying the symmetric equations B D#& % $

C"œ œ Þ

a) Do and ever intersect? Explain. _ `

b) Is parallel to ? Explain._ `

c) Is there an inconsistency in your answers to parts a) and b) ? Explain.

7. (1 point)Find an expression for a unit vector that bisects the angle between any two non-zero vectors and with ? + , + ,s s s Ä Ä

Á ! ÞÄ

8. (1 point) Let and be any two vectors.+ ,Ä Ä

a) Show that l ll l l l Ÿ l l Ÿ l l l lÄ Ä ÄÄ Ä Ä+ , + , + ,

b) Explain why is called the "Triangle Inequality".l l Ÿ l l l lÄ ÄÄ Ä+ , + ,

c) When does ?l l œ l l l lÄ ÄÄ Ä+ , + ,

d) When does l ll l l l œ l lÄ ÄÄ Ä+ , + , ?

Page 72 4

9. ( 4 points)Bonus Problem

Given that is an dimensional vector with each and , determine the choice of the EÄ

œ + ß + ß ß + 8 + ! + œ " +" # 8 4 4 4…4œ!

8

that minimizes the length of .EÄ

10. (2 points)a) Suppose that and are three mutually orthogonal unit vectors in , then any vector in can be resolved into iu v ws s , s d < dÄ3 3 tscomponents along the and directions. Determine a formula for the scalar components and whereu v ws s , s + , -ß ß

< œÄs+ , -u v ws s .

b) Given and , verify that these three vectors are mutually orthogonal unitu v ws s sœ œ œ"ß"ß " "ß #ß " "ß !ß"

$ ' #È È È , vectors. Here is shorthand for the vector . Bß Cß D DB C3s s s4 5

c) Given and ,u v ws s sœ œ œ"ß"ß " "ß #ß " "ß !ß"

$ ' #È È È , resolve into its components along the and directions.<Ä sœ $ß #ß $ u v ws s ,

d) Given and ,u v ws s sœ œ œ"ß"ß " "ß #ß " "ß !ß"

$ ' #È È È , resolve into its components along the and directions.< œ 4 5Ä sB C3s s s s s D u v w ,

Page 73 5

11. (1 point)

a) Express in terms of and l ‚E F E † F E † E † FÄ Ä Ä ÄÄ Ä Ä

lÄ# ß ÞF

b) Express in terms of , and Ð ‚ Ð ‚< F † E F † E < † F E † F F † FÄ ÄÄ Ä Ä Ä Ä ÄÑ Ñ

Ä Ä ÄÄ< ß Þ

12. (1 point)a) Given a point in the plane with coordinates and the line : determine an expression forT B C ÐB ß C Ñ C œ 7B ,: : _ ß

the distance from to .T _

b) Given a point with position vector and the line parallel to the vector and passing through the point with position veT VÄ Ä

: _ Z ctor

VÄ

! determine an expression for the distance from to .T _

c) Show that your answer to part b) gives the correct result when applied to the situation in part a).

Page 74 6

" Þ3 (2 points)Match each figure with an equation from the list below which could generate the surface shown. Write the answer in the blankbeneath the figure. a) b) c) d)C D œ " B C D œ ! B C D œ " B C D œ "# # # # # # # # # # # e) f) g) h) B C D œ ! B C D œ ! B C D œ " B C D œ "# # # # # # # # # #

i) j) k) l) B C D œ ! B C D œ ! B C œ " B D œ "# # # # # # # # # Figure A Figure B

Figure A Figure B________________________ ________________________

Figure C Figure D

Figure C Figure D________________________ ________________________

Page 75 7

Figure E Figure F

Figure E Figure F________________________ ________________________

Figure G Figure H

Figure G Figure H________________________ ________________________

Page 76 8

14. (2 points) Sketch the surface generated by each equation.

a) b) C D œ * B C D œ *# # # # #

c) d) B C D œ * B C D œ *# # # # # #

Page 77 9

15. (1 point)An artillery shell is fired from ground level at an angle of with respect to the horizontal. The muzzle velocity is per30° m(!! sec( ). Ignore air resistance and use a value of .1566 mph m per sec 1 œ *Þ)! 2

a) How long after it was fired does the shell hit the ground?

b) How far does the shell travel horizontally before it hits the ground?

In the following problems the position vector at time is ,the velocity vector is , and the acceleration vector is > Ä ÄV @ +Ä

16. ( 3 points)Bonus Problem Solve the following vector differential equation for : with V + 4 @ 4

ÄÐ>Ñ Ä Äœ Ð#>Ñ Ð!Ñ œ $sin s s s3 $#

17. (2 points)Simplify the following expressions :

a) dd>ŠV

Ä‚ @Ä ‹ œ __________________

b) dd>Š + @ÄÄ ‚ ‹ œ __________________

d) dd> @Ä

@± lÄ œ __________________

e) dd> @Ä

@± lÄ † @Ä œ __________________

Page 78 10

18. (1 point) Find expressions for the unit tangent vector and the unit normal vector to the curve at C œ ÐBÑ !cos Ð ß Ñ Þ1#

Ts œ _____________________

Rs œ _____________________

19. (1 point) Find the parametric equations for the tangent line to the given curve at the given point.

a) V 4 5Ä

Ð Ñ œ Ð#> "Ñ Ð$ &> Ñ Ð)> #Ñ> Ð "ß #ß "!Ñ $ #3s s s at Þ

b) V 4 5Ä

Ð Ñ œ / >/ Ð> 'Ñ> Ð "ß !ß 'Ñ > > #3s s s at Þ

20. ( ) Consider an object of constant mass, attracted to a force center by a force The angular momentum3 Bonus points m , . FÄ

is defined as

: fron Newton's second law: PÄ

œ ‚7 œÄ Ä V @ +Ä 7 F

Ä

a) Show that ddPÄ

> œ ‚Ä ÄV F

b) Suppose that is a central force. This means that where is a function only of the distance toF FÄ Ä

œ Ð ± ± ÑÄ

, f fVV

V

Ä

± ±Ä

the force center and not its direction. Show that must now be a constant vector.PÄ

c) Explain why this proves that in a central force the trajectory of an object is confined to the plane that contains the initialposition and velocity vectors.

Page 79 11





Page 80 Calculus II : Group Project 5 ( 8 points ) Due at Final Exam 5/12/17 Name _______________________ Name _______________________

Name _______________________ Name _______________________Choose two of the following six problems. All problems are worth 4 points. You may earn up to 20 bonus pointsby doing all problems.

1 . One of the five "Platonic solids" is the regular tetrahedron. This is a pyramid with four faces which are equilateral triangles. It hasfour vertices and six edges. Let be the length of each edge. Label the vertices as and Let be the centroid of triangle+ E F G H Sß ß Þ

EFG K I EF, the centroid of the solid tetrahedron, and the mid point of segment Þa) Determine the length ESÞ

b) Determine the height of the tetrahedron, i.e., determine the distance of from the plane containg andH E F G, Þ

c) Determine the distance of from the plane containg andK E F G, Þ

d) Determine the length EKÞ

e) Determine the to the nearest second.nSEH

f) Determine the to the nearest second.nSIH

g) Determine the to the nearest second.nSEK

h) Angle is important in organic chemistry where it is called the "tetrahedral bond angle". For example, the H C H bondEKH angle in the methane molecule is this angle. Determine the to the nearest second.nEKH


Name _______________________ Name _______________________

2. In cylindrical coordinates and one can form the following vectors :< ß ß D)

Y 3 4 Y 3 4s s s s< œ œ cos sin ; sin cos ;) ) ) )) a) Evaluate the following :

Y † Y œ Y † Y œ Y † 5 œ Y † Y œs< < < <________ ________ ________ _________) ) )

Y † 5 œ Y ‚Y œ Y ‚5 œ 5 ‚Y œs s s) ) )________ ________ ________ _________< <

b) The position vector can also be written as . What are the transformation equations forV œ 3 4 5 V œ Y 5s s s sp p

B C D < D<

the Cartesian coordinates and in terms of and ? Bß C D < ß ß Dß )

In spherical coordinates and one can form the following vectors :3 9 )ß ß

Y 3 4 5 Y 3 4 5s s s ss s3 9œ œ sin cos sin sin cos ; cos cos cos sin sin ;9 ) 9 ) 9 9 ) 9 ) 9

sin cos ; Y 3 4s s) œ ) )c) Evaluate the following :

Y † Y œ Y † Y œ Y † Y œ3 3 3 9 3 )__________ __________ __________

__________ __________ __________ Y † Y œ Y † Y œ Y † Y œ9 9 9 ) ) )

__________ __________ __________ Y ‚Y œ Y ‚Y œ Y ‚Y œ3 9 ) 3 9 )

d) The position vector can also be written as . What are the transformation equations for the V œ 3 4 5 V œ Ys s sp p

B C D 3 3 Cartesian coordinates and in terms of and ?Bß C D ßß 3 9 ),


Name _______________________ Name _______________________

3. For a planar curve the position vector from the origin to any point on the curve could be represented either in Cartesian<Ä coordinates as or in tangential-normal coordinates as , where and are the < 4 < sÄ Äœ œ X R T NB C3 X Rs s s tangential and normal coordinates, respectively.

a) Express and in terms of the Cartesian quantities and , ,T N .B C ddCB

X œ _____________________

R œ _____________________

b) Suppose the curve is expressed in polar form with as a function of Using the chain rule express in terms of the p< . ) ddCB olar

quantities and .< , ,) dd<)

ddCB œ _____________________

c) Express and in terms of the polar quantities , and X R < . , ) dd<)

T œ _____________________

N œ _____________________

The Federation Starship Exitprize has been captured by the evil Rhombulans from the planet Rhombus whose inhabitants would beconsidered squares if it weren't for their peculiar slant on life. A Rhombulian tractor beam is holding the Exitprize in a elliptical orbitwith Rhombus at one focus of the ellipse. Repeated efforts to escape have proven futile and have almost exhausted fuel supplies.Needless to say morale among the crew is alarmingly low and food is rapidly dwindling. By diverting all power to warp drive ChiefEngineer Mr. Scoot (who's not sure the engines can take it) says there is energy for just one more escape attempt. Captain Kwirk inconsultation with Science Officer and Calculus Expert Extraordinaire, Mr. Spook, discovers that due to a Rhombulian oversightescape is possible if the engines are fired at exactly the right position in the Exitprize's orbit.Specifically, if represents the planet Rhombulus at a focus of the ellipse and is the position of the ship on the ellipse, t F P hen the

vector can be written as , where and are unit tangent and normal vectors, respectively, to the ellipse. The FP T N Ä

T N T Ns ss s

engines must fire when the ratio is equal to the eccentricity of the ellipse. ± ±± ±TN

e) Letting Rhombus be the origin of a polar coordinate system and taking the polar angle to be zero when the Exitprize is at)perigee with Rhombus, at what angle(s) should the engines be fired?


Name _______________________ Name _______________________

4. A carnival ride called the "Monster" consists of a central arm of length which rotates at an angular velocity of rad5 m 0.35 ians persecond. Attached to the outer end of the central arm is a second arm of radius which rotates in the same direction with an2.5 mangular velocity of radians per second. A passenger carrying gondola is located at the outer end of this second arm.2.5

a) What is the maximum magnitude of the velocity in meters per second experienced by a person riding in the gondola?

b) What is the minimum magnitude of the velocity in meters per second experienced by a person riding in the gondola?

c) What is the maximum magnitude of the acceleration experienced by a person riding in the gondola? Express the answer inunits of per sec1 *Þ)! œ m 2Þ

d) What is the minimum magnitude of the acceleration experienced by a person riding in the gondola? Express the answer inunits of per sec1 *Þ)! œ m 2Þ

e) What is the time interval between the experience of maximum magnitude of acceleration and minimum magnitude ofacceleration?


Name _______________________ Name _______________________

5. A simple (though not very realistic) model for an object falling (the positive direction is up) through a viscous medium isD given bythe differential equation : , where is the acceleration due to gravity and is a positive constantm mg + 5 @ÄÄ œ s 1$ $(having units of mass per time) which reflects the effects of viscous drag.

a) Explain why it is important that the term be preceded by the minus sign.$

Consider an object governed by the above equation whose initial velocity is given by @Ä Ð Ñ œ @0 ! 3s .

b) Solve for the component of velocity, , as a function of time. ( is a constant.)B @ @B !

c) What happens to as gets big?@B >

d) Solve for as a function of @D > .

e) The combination is called the ' terminal velocity '. Explain this terminology.mg$ œ @>


Name _______________________ Name _______________________

6. What are the parametric equations for the coordinates of a particle on the rim of a wheel of radius mounted on aB Cß "

3 cm

wheel of radius mounted on a wheel of radius . The wheel of radius remains centered at the origin and rotates"# cm 1 cm 1 cm

counter clockwise with an angular veocity of radian per second. The second wheel of radius cm rotates counter1 "#

clockwise about its center at radians per second. The third wheel of radius rotates clockwise about its center at radi7 cm 17"3 ans

per second. The configuration at is as shown above on the right when the particle has coordinates > œ ! Ð ß Ñ Þ$ "# cm cm3

b) Using a computer plotting program graph the curve traced by these equations. Attach your graph with the project.

c) What symmetry does the curve traced by the particle exhibit?

d) Try to explain this symmetry. [ Hint : consider the complex function ]0Ð>Ñ œ BÐ>Ñ 3CÐ>Ñ Þ

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