tom robbins ww prob lib2 summer 2001 - math

Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Sequences1Definitions due 4/1/06 at 2:00 AM.

1.(1 pt) For each sequence, find a formula for the generalterm, an. For example, answer n2 if given the sequence:�

1 � 4 � 9 � 16 � 25 � 36 ��1. 1

2 � 14 � 1

8 � 116 ��

2. 12 � 2

3 � 34 � 4

5 ��

2.(1 pt) For each sequence, find a formula for the generalterm, An. For example, answer n2 if given the sequence:�

1 � 4 � 9 � 16 � 25 � 36 ��1. 3

16 � 425 � 5

36 � 649 ��

2. 12 � 1

4 � 16 � 1

8 ��

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�

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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Sequences2Limits due 4/2/06 at 1:00 AM.

1.(1 pt) Find the limit of the sequencean � 3n � 9

1n�

9

2.(1 pt) Find the limit of the sequence:an � 1n2 � 5n

�7

8n2 � 2n�

5

3.(1 pt) Find the limit of the sequence an � �cosn �5n �

4.(1 pt) Find the limit of the sequence whose terms are givenby

an �� n2 � � 1 � cos � 5 3n��

5.(1 pt) Determine whether the sequence is divergent or con-vergent. If it is convergent, evaluate its limit. If it diverges to in-finity, state your answer as ”INF” (without the quotation marks).If it diverges to negative infinity, state your answer as ”MINF”.If it diverges without being infinity or negative infinity, stateyour answer as ”DIV”.

limn � ∞

� 1n � 55n


limn � ∞

245n � 12arctan � n2 �


limn � ∞

18 � 5n � � 715 � 2n �


limn � ∞

� 6n5 � sin2 � 5n �n5 � 16

9.(1 pt) Determine whether the sequence is divergent or con-vergent. If it is convergent, evaluate its limit. If it diverges to in-finity, state your answer as ”INF” (without the quotation marks).

If it diverges to negative infinity, state your answer as ”MINF”.If it diverges without being infinity or negative infinity, stateyour answer as ”DIV”.

limn � ∞

20 � 24arctan � n! �2n

10.(1 pt) Determine whether the sequence is divergent orconvergent. If it is convergent, evaluate its limit. If it divergesto infinity, state your answer as ”INF” (without the quotationmarks). If it diverges to negative infinity, state your answer as”MINF”. If it diverges without being infinity or negative infin-ity, state your answer as ”DIV”.

limn � ∞

n3

e � 8n


limn � ∞

nn

e � 6n


limn � ∞

� 6 � n! �� 4 � n13.(1 pt) Determine whether the sequence is divergent or

convergent. If it is convergent, evaluate its limit. If it divergesto infinity, state your answer as ”INF” (without the quotationmarks). If it diverges to negative infinity, state your answer as”MINF”. If it diverges without being infinity or negative infin-ity, state your answer as ”DIV”.

limn � ∞

� 4 � n! �� n � n14.(1 pt) Find the limit of the sequence whose terms are given

byan �� e2n � 6n � 1 n �

15.(1 pt) Find the limit of the sequence whose terms are givenby

an � �1

e4n � n2 � 1 n �

16.(1 pt) If a sequence c1 � c2 � c3 �� has limit K then the se-quence ec1 � ec2 � ec3 �� has limit eK � Use this fact together withl’Hopital’s rule to compute the limit of the sequence given by

1

bn � � 1 � 2 4n� n �

17.(1 pt) If a sequence c1 � c2 � c3 �� has limit K then the se-quence ec1 � ec2 � ec3 �� has limit eK � Use this fact together withl’Hopital’s rule to compute the limit of the sequence given by

bn � � n � 2 � 5n �

18.(1 pt) Match each sequence below to statement that BESTfits it.STATEMENTSZ. The sequence converges to zero;I. The sequence diverges to infinity;F. The sequence has a finite non-zero limit;D. The sequence diverges.

SEQUENCES

1. n100�1 01 � n

2.�ln�n � �

n3. nsin � 1

n�

4. ln � ln � ln � n � ��5. n3 � 5n

3n � n5

6. n!n1000

7. arctan � n � 1 �8. sin � n �

19.(1 pt) Match each sequence below to statement that BESTfits it.STATEMENTSZ. The sequence converges to zero; I. The sequence diverges topositive infinity;F. The sequence has a finite non-zero limit; D. The sequencediverges, but not to infinity.

SEQUENCES

1.�

n2 � 4n � �n2

2. � 5n2n � 1 n3. 5n

n!4. � e

10� n

5.� � 5 � n

n!6. � � 1 � � n 2n

ln�n �

7. cos2 � n � � sin2 � n �8. 100n2 � 1

3n!


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Sequences3Monotone due 4/3/06 at 2:00 AM.

1.(1 pt) Determine whether the sequences are increasing, de-creasing, or not monotonic. If increasing, enter 1 as your an-swer. If decreasing, enter � 1as your answer. If not monotonic,enter 0 as your answer.

1. an � cosn4n

2. an � n � 4n�

4

3. an �� n�

46n�

4

4. an � 14n�

6

2.(1 pt) Let f � x � � xx2 � 4x � 41

�A. Find the smallest real number r such that f � x � is decreasingfor all x greater than r�

r �B. Find the smallest integer s such that f � n � is decreasing for

all integers n greater than or equal to s.s �


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Sequences4Arithmetic due 4/4/06 at 2:00 AM.

1.(1 pt) Write down the first five terms of the sequence�5n

n � 5 � , , , , ,

2.(1 pt) Find the 8th term of the arithmetic sequence� 5 �� 1 � 3 ��Answer:3.(1 pt) Find the sum

9 � 4 � 1 � �� 14 � 5n �Answer:4.(1 pt) Find the sum� 3 � 6 � 9 � �� 18Answer:5.(1 pt) Find the common difference and write out the first

four terms of the arithmetic sequence

�13

n � 34 �

Common difference isa1 � , a2 � , a3 � , a4 � ,6.(1 pt) Find the nth term of the arithmetic sequence whose

initial term is 6 and common difference is 3.(Your answer must be a function of n.)

7.(1 pt) Find the first term and the common difference of thearithmetic sequence whose 8th term is 42 and 13th term is 67.

First term is ,Common difference is

8.(1 pt) Find x such that 4x � 1 � 4x � 1 � and � 4x � 123 areconsecutive terms of an arithmetic sequence.

x �9.(1 pt) Write down the first five terms of the following re-

cursively defined sequence.

a1 � 2; an�

1 � � 2an � 1

, , , ,


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Sequences5Geometric due 4/5/06 at 2:00 AM.

1.(1 pt) Find the common ratio and write out the first four

terms of the geometric sequence

�3n�

3

8 �Common ratio isa1 � , a2 � , a3 � , a4 �2.(1 pt) Find the 4th term of the geometric sequence� 8 �� 40 �� 200 ��Answer:

3.(1 pt) Find the nth term of the geometric sequence whoseinitial term is 10 and common ration is 2.

(Your answer must be a function of n.)

4.(1 pt) Fred and Alice want to purchase a house. Supposethey invest 300 dollars per month into a mutual fund. How muchwill they have for a downpayment after 10 years if the per an-num rate of return of the mutual fund is assumed to be 9 � 5 per-cent compounded monthly?


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series1Definitions due 5/1/06 at 2:00 AM.

1.(1 pt) Let

sk � k

∑n � 1

n � � 1 � nFind s5 �s5 �

2.(1 pt) Let an be the n th digit after the decimal point in3π � 3e. Evaluate

∞

∑n � 1

an � � 1 � n �

3.(1 pt) Let r � 1327 �

For both of the following answer blanks, decide whether thegiven sequence or series is convergent or divergent. If conver-gent, enter the limit (for a sequence) or the sum (for a series). Ifdivergent, enter INF if it diverges to infinity, MINF if it divergesto minus infinity, or DIV otherwise.

A. Consider the sequence�nrn � .

limn � ∞ nrn �B. Take my word for it that it can be shown that

n

∑i � 1

iri � nrn�

2 � � n � 1 � rn�

1 � r� 1 � r � 2 �

Now consider the series ∑∞n � 1 nrn.

∑∞n � 1 nrn �4.(1 pt) Consider the series ∑∞

n � 16

n�

9 . Let sn be the n-th par-tial sum; that is,

sn � n

∑i � 1

6i � 9

�Find s4 and s8

s4 =s8 =

5.(1 pt)Two boys on bicycles, 67 miles apart, began racing directly

toward each other. The instant they started, a fly on the handlebar of one bicycle started flying straight toward the other cyclist.As soon as it reached the other handle bar it turned and startedback. The fly flew back and forth in this way, from handle barto handle bar, until the two bicycles met.

If each bicycle had a constant speed of 14 miles an hour, andthe fly flew at a constant speed of 20 miles an hour, how far didthe fly fly?

6.(1 pt) Evaluate the sum:4

∑k � 0

�� 1 � k � k � 1 � 2 � 7 �


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series2Telescope due 5/2/06 at 2:00 AM.

1.(1 pt) If the following series converges, compute its sum.Otherwise, enter INF if it diverges to infinity, MINF if it di-verges to minus infinity, and DIV otherwise.

∞

∑n � 1

9n � n � 2 �

(Hint: try breaking the summands up partial fractions-style.)2.(1 pt) For the following series, if it converges, enter the

limit of convergence. If not, enter ”DIV” (unquoted).∞

∑n � 1

ln � 2 � n � 1 � � � ln � 2n �3.(1 pt) Determine the sum of the following series.

∞

∑n � 1� sin � 7n � � sin � 7

n � 1� �

4.(1 pt) Decide whether each of the following series con-verges. If a given series converges, compute its sum. Otherwise,

enter INF if it diverges to infinity, MINF if it diverges to minusinfinity, and DIV otherwise.

1.∞

∑n � 1� sin � 10

n� � sin � 10

n � 1� �

2.∞

∑n � 1� sin � 10n � � sin � 10 � n � 1 � ��

3.∞

∑n � 1� e � 9n � � e9

�n�

1 � �5.(1 pt) If the following series converges, compute its sum.

Otherwise, enter INF if it diverges to infinity, MINF if it di-verges to minus infinity, and DIV otherwise.

∞

∑n � 1� e � � 9n � � e � 9

�n�

1 � �


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series3Convergent due 5/3/06 at 2:00 AM.

1.(1 pt) Given:An � 5n� 2n

�3

For both of the following answer blanks, decide whether thegiven sequence or series is convergent or divergent. If conver-gent, enter the limit (for a sequence) or the sum (for a series). Ifdivergent, enter INF if it diverges to infinity, MINF if it divergesto minus infinity, or DIV otherwise.

(a) The series∞

∑n � 1� An� .

(b) The sequence�An � .

2.(1 pt) Given:

An � 4096n�2

82n

Determine: (a) whether∞

∑n � 1� An� is convergent.

(b) whether�An � is convergent. If convergent, enter the

limit of convergence. If not, enter ”DIV” (unquoted).

3.(1 pt) Given:An � 20

2n

Determine:

(a) whether∞

∑n � 1� An� is convergent. and

(b) whether�An � is convergent.

If convergent, enter the limit of convergence. If not, enter”DIV” (unquoted).

4.(1 pt) Given:An � 8n

80


∑n � 1� An� is convergent. (b)

whether�An � is convergent. If convergent, enter the limit

of convergence. If not, enter ’DIV’ (unquoted).

5.(1 pt) Match each of the following with the correct state-ment.C stands for Convergent, D stands for Divergent.

1.∞

∑n � 1

5n4 � 9

2.∞

∑n � 1

5n � n � 4 �

3.∞

∑n � 1

6 � 1n

9 � 8n

4.∞

∑n � 1

1

3 � 2�

n4

5.∞

∑n � 1

ln � n �4n


1.∞

∑n � 1

6 � 6n

7 � 5n

2.∞

∑n � 1

3n4 � 4

3.∞

∑n � 1

1

2 � 4�

n5

4.∞

∑n � 1

ln � n �4n

5.∞

∑n � 1

3n � n � 5 �

7.(1 pt) Determine whether the series is convergent or diver-gent.

∞

∑n � 1� � 5

2�

n5� 6

n5�

If convergent, enter the 5 th partial sum to estimate the sumof the series; otherwise, enter DIV.(Note: if you have trouble reading this problem, try selectingtypeset mode below and then hitting the submit answer button.)


1.∞

∑n � 1

6 � 10n

1n

2.∞

∑n � 1

n7

n7 � 2

3.∞

∑n � 1

ne � n2

4.∞

∑n � 2

310n ln � n �

5.∞

∑n � 1

4n7 � n10


1.∞

∑n � 2

102n ln � n �

2.∞

∑n � 1

1 � 7n

8n

3.∞

∑n � 1

ne � n2

4.∞

∑n � 1

n5

n7 � 6

5.∞

∑n � 1

5n7 � n2

1

10.(1 pt) For each of the following series, tell whether or notyou can apply the 3-condition test (i.e. the alternating seriestest). If you can apply this test, enter D if the series diverges,or C if the series converges. If you can’t apply this test (even ifyou know how the series behaves by some other test), enter N.

1.∞

∑n � 1

� � 1 � nn5

2.∞

∑n � 1

� � 1 � n cos � n �n2

3.∞

∑n � 1

� � 1 � n � n4 � 2n �n3 � 1

4.∞

∑n � 1

� � 1 � n � n3 � 1 �n3 � 7

5.∞

∑n � 1

� � 1 � n � n10 � 1 �en

6.∞

∑n � 1

� � 1 � n � n3 � 1 �n4 � 1

11.(1 pt) For the following alternating series,∞

∑n � 1

an � 1 � 110� 1

100� 1

1000� ��

how many terms do you have to go for your approximation(your partial sum) to be within 1e-09 from the convergent valueof that series?

12.(1 pt) For each of the following series, tell whether or notyou can apply the 3-condition test (i.e. the alternating seriestest). If you can apply this test, enter D if the series diverges,or C if the series converges. If you can’t apply this test (even ifyou know how the series behaves by some other test), enter N.

1.∞

∑n � 1

� � 1 � nn!nn

2.∞

∑n � 1

� � 1 � nnn

n!

3.∞

∑n � 1

� � 1 � nn!en

4.∞

∑n � 1

� � 1 � ncos � nπ �n5

5.∞

∑n � 1

� � 1 � nn2 � 5

6.∞

∑n � 1

� � 1 � nen

n!

13.(1 pt) If the following series converges, compute its sum.Otherwise, enter INF if it diverges to infinity, MINF if it di-verges to minus infinity, and DIV otherwise.

∞

∑n � 1

3 � 9n

9n


∑n � 1

an � 0 � 4 � � 0 � 4 � 33!

� � 0 � 4 � 55!

� � 0 � 4 � 77!

� ��how many terms do you have to go for your approximation

(your partial sum) to be within 0.0000001 from the convergentvalue of that series?


∑n � 1

an � 1 � � 0 � 35 � 22!

� � 0 � 35 � 44!

� � 0 � 35 � 66!

� � 0 � 35 � 88!

� ��how many terms do you have to go for your approximation

(your partial sum) to be within 0.0000001 from the convergentvalue of that series?


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series4Geometric due 5/4/06 at 2:00 AM.

1.(1 pt) Determine the sum of the following series.∞

∑n � 1

� � 3 � n � 1

6n


∑n � 1

� � 3 � n � 1

5n


∑n � 1� 2n � 5n

8n�

4.(1 pt) Determine the sum of the following series by com-puting the first 10 partial sums.

∞

∑n � 1� 603n�

5.(1 pt) Given:An � 2n

9n�

7


∑n � 1� An� is convergent.

(b) whether�An � is convergent. If convergent, enter the

limit of convergence. If not, enter ”divergent” (unquoted).

6.(1 pt) Express 4 � 63636363636 �� as a rational number, inthe form p

qwhere p and q are positive integers with no common factors.p = andq =

7.(1 pt) Express 5 � 81818181818 �� as a rational number, inthe form p

qwhere p and q are positive integers with no common factors.p = andq =

8.(1 pt) Express 7 � 603603603 �� as a rational number, in theform p

qwhere p and q have no common factors.p = andq =

9.(1 pt)153846153846...The start of an infinite repeating list of digits is given above.A new list is made by discarding the first 3 digits from the infi-nite list above.Let m be the six digit integer formed by the first six digits of thenew list.Let r be the number given by the decimal obtained by putting adecimal pointat the start of the new infinite list. The number r is rationaland can be written as a fraction p/q,

where p and q are positive integers and have no common factorgreater than one.Find p and q.(The integer m is a multiple of 999.)p =

q =

10.(1 pt) Let r � 1626 � It can be shown that

� ln � 1 � r � � ∞

∑n � 1

1n

rn �

Let

sk � k

∑1

1n

rn �

A.Find the smallest number M such that sk�

M for every posi-tive integer k �M �B.Find s3 �s3 �

C. Note that 1 � r � 1026 � Then � ln � 1 � r � � ln � 26

10� �

Suppose s3 is used to approximate ln � 2610� �

The error is∞

∑n � 4� 1n � rn � which is less than

14

∞

∑n � 4

rn �Use the formula for the sum of a geometric seriesto calculate this last sum and thereby to estimate the error in theapproximation.ERROR

�Your answer to C. should be more than the actual error which is0.0730960300676426.

11.(1 pt) The geometric series can be used to approximatethe reciprocal of a number by using a nearby number whose re-

ciprocal is known. For example,1

24 � 125

1

1 � 125

=

125

∞

∑n � 0

�125 � n

leads to the approximation 125 � 1 � 1

25� of 1

24 by truncating theseries.This approximation to 1

24 is easily expressed as a decimal:.04(1 +.04) = .0416.

Use the fact that 110 is near 100 to get a similar four placedecimal approximation of 1

110 �

The error in approximating a number A by a number a is e= a-A. The relative error is e/A. The relative percent error is100e/A.Find the relative percent error in the approximation of 1

110 de-scribed above.

12.(1 pt) A ball drops from a height of 12 feet. Each time ithits the ground, it bounces up 25 percents of the height it fall.Assume it goes on forever, find the total distance it travels.

1

13.(1 pt) Find the sum� 8 � � 8

3 � � 89 � �� 8

3n � 1

Answer:


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series5IntegralTest due 5/5/06 at 2:00 AM.

1.(1 pt) Compute the value of the following improper integralif it converges. If it diverges, enter INF if it diverges to infinity,MINF if it diverges to minus infinity, or DIV otherwise (hint:integrate by parts).

� ∞

1

3ln � x �x2 dx

Determine whether∞

∑n � 1� 3ln � n �

n2�

is a convergent series. Enter C if the series is convergent, or Dif it is divergent.

2.(1 pt) Find the value of� ∞

2

dx4x � ln � 9x � � 2


∑n � 2� 14n � ln � 9n �� 2 �

Enter A if series is convergent, B if series is divergent.3.(1 pt) Find the value of� ∞

2

dx� 3x � 2 � 2Determine whether

∞

∑n � 2� 1� 3x � 2 � 2 �

Enter A if series is convergent, B if series is divergent.

4.(1 pt) � a � Compute s6 (the 6th partial sum) of s � ∞

∑n � 1

47n5

� b � Estimate the error in using s6 as an approximation of the

sum of the series. (i.e. use� ∞

6f � x � dx � R6 )

� c � Use n = 6 and

sn � � ∞

n�

1f � x � dx

�s

�sn � � ∞

nf � x � dx

to find a better estimate of the sum.�s

�5.(1 pt) Test each of the following series for convergence by

the Integral Test. If the Integral Test can be applied to the series,enter CONV if it converges or DIV if it diverges. If the integraltest cannot be applied to the series, enter NA. (Note: this meansthat even if you know a given series converges by some othertest, but the Integral Test cannot be applied to it, then you mustenter NA rather than CONV.)

1.∞

∑n � 1

ne � 7n

2.∞

∑n � 1

3n ln � 3n �

3.∞

∑n � 1

ln � 9n �n

4.∞

∑n � 1

n � 7� � 4 � n5.

∞

∑n � 1

ne7n


1

5dxx2 � 1


∑n � 1� 5n2 � 1

�Enter A if series is convergent, B if series is divergent.


12x2e � x3


∑n � 1� 2n2e � n3

Enter A if series is convergent, B if series is divergent.


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series6CompTests due 5/6/06 at 2:00 AM.

1.(1 pt) Test each of the following series for convergence byeither the Comparison Test or the Limit Comparison Test. If ei-ther test can be applied to the series, enter CONV if it convergesor DIV if it diverges. If neither test can be applied to the series,enter NA. (Note: this means that even if you know a given seriesconverges by some other test, but the comparison tests cannot beapplied to it, then you must enter NA rather than CONV.)

1.∞

∑n � 1

cos2 � n � �n

n6

2.∞

∑n � 1

� ln � n � � 2n � 4

3.∞

∑n � 1

9n6

n10 � 3

4.∞

∑n � 1

9n6

n7 � 3

5.∞

∑n � 1

� � 1 � n6n

2.(1 pt) Test each of the following series for convergence byeither the Comparison Test or the Limit Comparison Test. If ei-ther test can be applied to the series, enter CONV if it convergesor DIV if it diverges. If neither test can be applied to the series,enter NA. (Note: this means that even if you know a given seriesconverges by some other test, but the comparison tests cannot beapplied to it, then you must enter NA rather than CONV.)

1.∞

∑n � 1

9n5

n8 � 3

2.∞

∑n � 1

� ln � n � � 4n � 5

3.∞

∑n � 1

cos2 � n � �n

n5

4.∞

∑n � 1

5n8 � n6 � 9�

n7n10 � n5 � 4

5.∞

∑n � 1

cos � n � �n

9n � 3

3.(1 pt) Test each of the following series for convergence byeither the Comparison Test or the Limit Comparison Test. Ifat least one test can be applied to the series, enter CONV if itconverges or DIV if it diverges. If neither test can be applied tothe series, enter NA. (Note: this means that even if you knowa given series converges by some other test, but the comparisontests cannot be applied to it, then you must enter NA rather thanCONV.)

1.∞

∑n � 1

cos � n � �n

9n � 4

2.∞

∑n � 1

� � 1 � n8n

3.∞

∑n � 1

� ln � n �� 4n � 8

4.∞

∑n � 1

cos2 � n � �n

n4

5.∞

∑n � 1

9n4

n9 � 4

4.(1 pt) Test each of the following series for convergence byeither the Comparison Test or the Limit Comparison Test. Ifat least one test can be applied to the series, enter CONV if itconverges or DIV if it diverges. If neither test can be applied tothe series, enter NA. (Note: this means that even if you knowa given series converges by some other test, but the comparisontests cannot be applied to it, then you must enter NA rather thanCONV.)

1.∞

∑n � 1

3n5

n9 � 4

2.∞

∑n � 1

� ln � n �� 3n � 6

3.∞

∑n � 1

3n5

n6 � 4

4.∞

∑n � 1

cos2 � n � �n

n5

5.∞

∑n � 1

� � 1 � n2n

5.(1 pt) Each of the following statements is an attempt toshow that a given series is convergent or not using the Compari-son Test (NOT the Limit Comparison Test.) For each statement,enter C (for ”correct”) if the argument is valid, or enter I (for”incorrect”) if any part of the argument is flawed. (Note: if theconclusion is true but the argument that led to it was wrong, youmust enter I.)

1. For all n � 2, ln�n �

n � 1n , and the series ∑ 1

n diverges, so

by the Comparison Test, the series ∑ ln�n �

n diverges.2. For all n � 1, n

4 � n3� 1

n2 , and the series ∑ 1n2 converges,

so by the Comparison Test, the series ∑ n4 � n3 converges.

3. For all n � 1, 1n ln�n � � 2

n , and the series 2∑ 1n diverges,

so by the Comparison Test, the series ∑ 1n ln�n � diverges.

4. For all n � 2, nn3 � 7

� 2n2 , and the series 2∑ 1

n2 con-verges, so by the Comparison Test, the series ∑ n

n3 � 7converges.

5. For all n � 1, ln�n �

n2� 1

n1 � 5 , and the series ∑ 1n1 � 5 con-

verges, so by the Comparison Test, the series ∑ ln�n �

n2

converges.6. For all n � 2, 1

n2 � 7� 1

n2 , and the series ∑ 1n2 converges,

so by the Comparison Test, the series ∑ 1n2 � 7

converges.1

6.(1 pt) Each of the following statements is an attempt toshow that a given series is convergent or not using the Compari-son Test (NOT the Limit Comparison Test.) For each statement,enter C (for ”correct”) if the argument is valid, or enter I (for”incorrect”) if any part of the argument is flawed. (Note: if theconclusion is true but the argument that led to it was wrong, youmust enter I.)

1. For all n � 2,ln � n �

n�

1n

, and the series ∑ 1n

diverges,

so by the Comparison Test, the series ∑ ln � n �n

diverges.

2. For all n � 1,arctan � n �

n3� π

2n3 , and the seriesπ2 ∑ 1

n3

converges, so by the Comparison Test, the series

∑ arctan � n �n3 converges.


n2 �1n2 , and the series ∑ 1

n2 con-

verges, so by the Comparison Test, the series ∑ ln � n �n2

converges.

4. For all n � 2,n

n3 � 6� 2

n2 , and the series 2∑ 1n2 con-

verges, so by the Comparison Test, the series ∑ nn3 � 6

converges.

5. For all n � 2,1

n2 � 7� 1

n2 , and the series ∑ 1n2 con-

verges, so by the Comparison Test, the series ∑ 1n2 � 7

converges.


n2� 1

n1 5 , and the series ∑ 1n1 5 con-

verges, so by the Comparison Test, the series ∑ ln � n �n2

converges.

7.(1 pt) The three series ∑An, ∑Bn, and ∑Cn have terms

An � 1n9 � Bn � 1

n5 � Cn � 1n

�Use the Limit Comparison Test to compare the following seriesto any of the above series. For each of the series below, you mustenter two letters. The first is the letter (A,B, or C) of the seriesabove that it can be legally compared to with the Limit Compar-ison Test. The second is C if the given series converges, or D ifit diverges. So for instance, if you believe the series convergesand can be compared with series C above, you would enter CC;or if you believe it diverges and can be compared with series A,you would enter AD.

1.∞

∑n � 1

3n5 � n2 � 3n12n14 � 8n11 � 7

2.∞

∑n � 1

8n2 � 3n8

5n9 � 12n3 � 5

3.∞

∑n � 1

5n5 � n9

1309n14 � 12n5 � 8

8.(1 pt) The three series ∑An, ∑Bn, and ∑Cn have terms

An � 1n7 � Bn � 1

n4 � Cn � 1n

�Use the Limit Comparison Test to compare the following seriesto any of the above series. For each of the series below, you mustenter two letters. The first is the letter (A,B, or C) of the seriesabove that it can be legally compared to with the Limit Compar-ison Test. The second is C if the given series converges, or D ifit diverges. So for instance, if you believe the series convergesand can be compared with series C above, you would enter CC;or if you believe it diverges and can be compared with series A,you would enter AD.

1.∞

∑n � 1

4n4 � n7

1122n11 � 5n4 � 2

2.∞

∑n � 1

2n2 � 3n6

4n7 � 5n3 � 4

3.∞

∑n � 1

3n4 � n2 � 3n5n11 � 2n9 � 6

9.(1 pt) Select the FIRST correct reason why the given seriesconverges.

A. Convergent geometric seriesB. Convergent p seriesC. Comparison (or Limit Comparison) with a geometric or

p seriesD. Converges by alternating series test

1.∞

∑n � 1

� � 1 � n ln � en �n8 cos � nπ �

2.∞

∑n � 1

cos � nπ �ln � 6n �

3.∞

∑n � 1� � 1 � n �

nn � 1

4.∞

∑n � 1

sin2 � 6n �n2

5.∞

∑n � 1

n2 � �n

n4 � 1

6.∞

∑n � 1

� � 1 � n6n � 6

10.(1 pt) Select the FIRST correct reason why the given se-ries converges.

A. Convergent geometric seriesB. Convergent p seriesC. Comparison (or Limit Comparison) with a geometric or

p seriesD. Cannot apply any test done so far in class

1.∞

∑n � 1

cos � nπ �ln � 2n �

2.∞

∑n � 1� � 1 � n �

nn � 5

2

3.∞

∑n � 1

6 � 7 � n112n

4.∞

∑n � 1

n2 � �n

n4 � 5

5.∞

∑n � 1

� � 1 � n ln � en �n7 cos � nπ �

6.∞

∑n � 1

� n � 1 � � 3 � n22n

11.(1 pt) Select the FIRST correct reason why the given se-ries converges.

A. Convergent geometric seriesB. Convergent p seriesC. Integral testD. Comparison with a convergent p seriesE. Converges by limit comparison testF. Converges by alternating series test

1.∞

∑n � 1

sin2 � 4n �n2

2.∞

∑n � 1

n2 � �n

n4 � 7

3.∞

∑n � 1

� � 1 � n4n � 7

4.∞

∑n � 1

� n � 1 � � 42 � 1 � n42n

5.∞

∑n � 1

� cos � nπ �ln � 4n �

6.∞

∑n � 1� � e

π� n

12.(1 pt) Select the FIRST correct reason why the given se-ries diverges.

A. Diverges because the terms don’t have limit zeroB. Divergent geometric seriesC. Divergent p seriesD. Integral testE. Comparison with a divergent p seriesF. Diverges by limit comparison test

G. Diverges by alternating series test

1.∞

∑n � 1

ln � n �n

2.∞

∑n � 1

� n � 1 � � 42 � 1 � n42n

3.∞

∑n � 1� � 1 � n � 2n � !� n! � 2

4.∞

∑n � 1

5n � 3� � 1 � n5.

∞

∑n � 1� n � � 1

6

6.∞

∑n � 1

1�n

13.(1 pt) A. Suppose that f(x) is a function that is positiveand decreasing. Recall that by the integral test:� ∞

pf � x � dx

� ∞

∑n � p

f � n � �Recall that e � ∞

∑n � 0

1n!

� Suppose that for each positive integer k,

f � k � � 1k! . Find an upper bound B for� ∞

2f � x � dx �

B =B. A function is given by

h � k � � � ∞

0xke � xdx �

. Its values may be found in tables. Make the change of vari-ables y � x ln � 5 � to express

I � � ∞

0x45 � xdx as a constant C times h � 4 � � Find C.

C =C. Let g � x � � x45 � n � Find the smallest number M such that

the function g is decreasing for all x � M �C. M =

D. Does∞

∑n � 1

n45 � n converge or diverge?

Answer with one letter, C or D.

14.(1 pt) For each sequence an find a number k such that nkan

has a finite non-zero limit.(This is of use, because by the limit comparison test the series

∞

∑n � 1

an and∞

∑n � 1

n � k both converge or both diverge.)

A. an � � 7 � 6n � � 3

k =B. an � 3

n6 � nk =C. an � 2n2 � 3n

�2

6n6 � 6n�

6k =

D. an � �2n2 � 3n

�7

6n6 � 6n�

6 � n � 9

k =15.(1 pt) For each sequence an find a number r such that an

rn

has a finite non-zero limit.(This is of use, because by the limit comparison test the series

∞

∑n � 1

an and∞

∑n � 1

rn both converge or both diverge.)

A. an � � 4 � 3n � � 5

r =B. an � 52n

7n � nr =C. an � 7n � n5 � 7

152n � 7n � 3r =

D. an � �7n2 � 5n

�40 � 2n

15n � 2 � 7n�

3 � n � 7

3

r =

16.(1 pt) The series∞

∑n � 1

nkrn converges when 0 � r � 1 and

diverges when r � 1 � This is true regardless of the value of theconstant k . When r � 1 the series is a p-series. It converges ifk

� � 1and diverges otherwise. Each of the series below can be

compared to a series of the form∞

∑n � 1

nkrn � For each series deter-

mine the best value of r and decide whether the series converges.

A.∞

∑n � 1

� 4 � n � 4 � n � � 2

r = converges or diverges (c or d)?

B.∞

∑n � 1

nπ 22n

6n � n9


C.∞

∑n � 1

n 5 � 5n6 � 4


D.∞

∑n � 1

�5n2 � 2n � 4 � 4n

5n�

4 � 6n � 4�

n � 4


17.(1 pt) For each of the series below select the letter from ato c that best applies and the letter from d to k that best applies.A possible answer is af, for example.

A. The series is absolutely convergent.B. The series converges, but not absolutely.C. The series diverges.D. The alternating series test shows the series converges.E. The series is a p-series.F. The series is a geometric series.

G. We can decide whether this series converges by com-parison with a p series.

H. We can decide whether this series converges by com-parison with a geometric series.

I. Partial sums of the series telescope.J. The terms of the series do not have limit zero.

K. None of the above reasons applies to the convergenceor divergence of the series.

1.∞

∑n � 1

4 � sin � n ��n

2.∞

∑n � 2

1n log � 5 � n �

3.∞

∑n � 1

cos2 � nπ �nπ

4.∞

∑n � 1

� 2n � 2 � !� n! � 25.

∞

∑n � 1

1n

�n

6.∞

∑n � 1

cos � nπ �nπ

18.(1 pt)For each of the series below select the letter from a to c that bestapplies and the letter from d to k that best applies. A possiblecorrect answer is af, for example.

A. The series is absolutely convergent.B. The series converges, but not absolutely.C. The series diverges.D. The alternating series test shows the series converges.E. The series is a p-series.F. The series is a geometric series.

G. We can decide whether this series converges by com-parison with a p series.

H. We can decide whether this series converges by com-parison with a geometric series.

I. Partial sums of the series telescope.J. The terms of the series do not have limit zero.

1.∞

∑n � 1� log � n � 1 � � logn �

2.∞

∑n � 1

nn � 5n

3.∞

∑n � 1

n4

4n

4.∞

∑n � 5

� 7 � 1 �� 2 � 7 � 1 �� n � 1 � 7 � 1 �7n � n! � �

n

5.∞

∑n � 1

�1 � 4

n � n

6.∞

∑n � 1� � 1 � n � n

�1

n5 � xdx


A. Diverges because the terms don’t have limit zeroB. Divergent geometric seriesC. Divergent p seriesD. Integral testE. Comparison with a divergent p seriesF. Diverges by limit comparison test

G. Cannot apply any test done so far in class

1.∞

∑n � 1

cos � nπ �ln � 5 �

2.∞

∑n � 1

1n ln � n �

3.∞

∑n � 1

7n � 4� � 1 � n4.

∞

∑n � 1

� n � 1 � � 82 � 1 � n82n

5.∞

∑n � 1� � 1 � n � 2n � !� n! � 2

6.∞

∑n � 1

ln � n �n

4

Here is a short review of numerical series which you may findhelpful.REVIEW OF NUMERICAL SERIESSEQUENCESA sequence is a list of real numbers. It is called convergent if ithas a limit. An increasing sequence has a limit when it has anupper bound.

SERIES(Geometric series,rational numbers as decimals, harmonic se-ries,divergence test)Given numbers forming a sequence a1 � a2 �� let us define thenth partial sum as sum of the first n of them sn � a1 � �� an �The SERIES is convergent if the SEQUENCE s1 � s2 � s3 �� is.In other words it converges if the partial sums of the series ap-proach a limit.A necessary condition for the convergence of this SERIES isthat a’s have limit 0. If this fails, the series diverges.The harmonic series 1+(1/2)+(1/3)+... diverges.This illustrates that the terms an having limit zero does not guar-antee the convergence of a series.A series with positive terms ,i.e. an � 0 for all n, convergesexactly when its partial sums have an upper bound.

The geometric series∞

∑n � 1

rn converges exactly when � 1 �r

� 1 �INTEGRAL AND COMPARISON TESTS(Integral test,p-series, comparison tests for convergence and di-vergence, limit comparison test)Integral test: Suppose f � x � is positive and DECREASING forall large enough x. Then the following are equivalent:

I.� ∞

1f � x � dx is finite, i.e. converges.

S.∞

∑n � 1

f � n � is finite, i.e. converges.

This gives the p - test:∞

∑n � 1

1np converges exactly when p � 1 �

Comparison test: Suppose there is a fixed number K such thatfor all sufficiently large n: 0 � an

� Kbn �Convergence. If

∞

∑n � 1

bn converges then so does∞

∑n � 1

an �

Divergence. If∞

∑n � 1

andiverges then so does∞

∑n � 1

bn.

(Positive series having smaller terms are more likely to con-verge.)

Limit comparison test: SUPPOSE: an � 0, bn � 0 and

limn � ∞

an

bn� R exists. Moreover, R is not zero.

THEN∞

∑n � 1

an and∞

∑n � 1

bn

both converge or both diverge.OTHER CONVERGENCE TESTS FOR SERIES

(Alternating series test, absolute convergence, RATIO TEST)Alternating series test: Suppose the sequence a1 � a2 � a3 �� is de-

creasing and has limit zero. Then∞

∑n � 1

� � 1 � nan converges.

This applies to (1)-(1/2)+(1/3)-(1/4)+...

Absolute Convergence Test: IF∞

∑n � 1

�an

�converges,

THEN∞

∑n � 1

an converges.

Ratio test:SUPPOSE

� an � 1an

�has limit equal to r.

IF r� 1 then

∞

∑n � 1

an CONVERGES.

IF r � 1 the∞

∑n � 1

an DIVERGES.


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5

Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series7AbsolutelyConvergent due 5/7/06 at 2:00 AM.

1.(1 pt) Match each of the following with the correct state-ment.A. The series is absolutely convergent.C. The series converges, but is not absolutely convergent.D. The series diverges.

1.∞

∑n � 1� � 1 � n �

nn � 3

2.∞

∑n � 1

sin � 4n �n2

3.∞

∑n � 1

� � 1 � n3n � 6

4.∞

∑n � 1

� n � 1 � � 22 � 1 � n22n

5.∞

∑n � 1

� � 6 � nn7


1.∞

∑n � 1

sin � 6n �n2

2.∞

∑n � 1

� n � 1 � � 22 � 1 � n22n

3.∞

∑n � 1

� � 1 � n4n � 4

4.∞

∑n � 1

� � 4 � nn6

5.∞

∑n � 1� � 1 � n �

nn � 6


1.∞

∑n � 1

� n � 2 � !10nn!

2.∞

∑n � 1

� � 1 � n6n � 1

� 6 � n � 1n15

3.∞

∑n � 1� � 1 � n n!

4n

4.∞

∑n � 1

� � 1 � n4nn!

5.∞

∑n � 1

n5

5n


1.∞

∑n � 1

� � 1 � n2n � 1

� 2 � n � 1n15

2.∞

∑n � 1

� � 1 � n5nn!

3.∞

∑n � 1

n6

5n

4.∞

∑n � 1

� n � 2 � !2nn!

5.∞

∑n � 1� � 1 � n n!

5n


1.∞

∑n � 1

� � 1 � n � 1

2n � 4

2.∞

∑n � 1

� n � 5 � !n!2n

3.∞

∑n � 1� � 1 � n � 1 � 9 � n � 3n

� n2 � 52n

4.∞

∑n � 1

sin � 3n �n3

5.∞

∑n � 1

� � 3 � nn5


1.∞

∑n � 1

� � 1 � n � 1

2n � 1

2.∞

∑n � 1� � 1 � n � 1 � 9 � n � 2n

� n2 � 42n

3.∞

∑n � 1

� � 3 � nn4

4.∞

∑n � 1

� n � 4 � !n!4n

5.∞

∑n � 1

sin � 3n �n5

1


1.∞

∑n � 1

� n � 3 � n4n

2.∞

∑n � 1� � 1 � n � 1 � 10 � n � 5n

� n2 � 42n

3.∞

∑n � 1

� � 4n � nn6n

4.∞

∑n � 1

�n6

5 � 8n4 � n

5.∞

∑n � 1

� n � 3 � n4n2

6.∞

∑n � 1� � 1 � nn � 2 ln � n � 6 �

8.(1 pt) Consider the series∞

∑n � 1

an where

an � � � 8 � n � 1

� 3n � 5 � 10n

In this problem you must attempt to use the Ratio Test to decidewhether the series converges.

Compute

L � limn � ∞

��an�

1

an

��

Enter the numerical value of the limit L if it converges, INF ifit diverges to infinity, MINF if it diverges to negative infinity, orDIV if it diverges but not to infinity or negative infinity.L �

Which of the following statements is true?A. The Ratio Test says that the series converges absolutely.B. The Ratio Test says that the series diverges.C. The Ratio Test says that the series converges conditionally.D. The Ratio Test is inconclusive, but the series converges ab-solutely by another test or tests.E. The Ratio Test is inconclusive, but the series diverges by an-other test or tests.F. The Ratio Test is inconclusive, but the series converges con-ditionally by another test or tests.Enter the letter for your choice here:


∑n � 1

an where

an � en�

3�

n � 6� n � 5 � !In this problem you must attempt to use the Ratio Test to decidewhether the series converges.

Compute

L � limn � ∞

��an�

1

an

��




∑n � 1

an where

an � nn2 � 5n � 4

In this problem you must attempt to use the Ratio Test to decidewhether the series converges.

Compute

L � limn � ∞

��an�

1

an

��




∑n � 1

an where

an � � 2n � 1 � 2n

� n � 7 � 2n

In this problem you must attempt to use the Root Test to decidewhether the series converges.

Compute

L � limn � ∞

n� �

an�

2


Which of the following statements is true?A. The Root Test says that the series converges absolutely.B. The Root Test says that the series diverges.C. The Root Test says that the series converges conditionally.D. The Root Test is inconclusive, but the series converges abso-lutely by another test or tests.E. The Root Test is inconclusive, but the series diverges by an-other test or tests.F. The Root Test is inconclusive, but the series converges condi-tionally by another test or tests.Enter the letter for your choice here:

12.(1 pt) Consider the series ∑∞n � 1 an where

an �� 1 � n � ln � n �n� n

In this problem you must attempt to use the Root Test to decidewhether the series converges.

ComputeL � lim

n � ∞n

� �an

�


Which of the following statements is true?A. The Root Test says that the series converges absolutely.B. The Root Test says that the series diverges.C. The Root Test says that the series converges conditionally.D. The Root Test is inconclusive, but the series converges abso-lutely by another test or tests.E. The Root Test is inconclusive, but the series diverges by an-other test or tests.F. The Root Test is inconclusive, but the series converges condi-tionally by another test or tests.Enter the letter for your choice here:


UR

3

Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series8Power due 5/8/06 at 2:00 AM.

1.(1 pt) Find the interval of convergence for the given powerseries.

∞

∑n � 1

� x � 11 � nn � � 9 � n

The series is convergentfrom x = , left end included (Y,N):to x = , right end included(Y,N):

2.(1 pt) Find all the values of x such that the given serieswould converge.

∞

∑n � 1

� 3x � nn10

The series is convergentfrom x = , left end included (Y,N):

to x = , right end included(Y,N):3.(1 pt) Find all the values of x such that the given series

would converge.

∞

∑n � 1

� x � 2 � n� 2 � nThe series is convergent

from x = , left end included (Y,N):to x = , right end included(Y,N):


∞

∑n � 1

� � 1 � nxn

� 8 � n � n2 � 3 �The series is convergentfrom x = , left end included (Y,N):


would converge.

∞

∑n � 1

� � 1 � n � 9 � nxn

� �n � 9 �



would converge.

∞

∑n � 1

xn

� 2 � n � �n � 2 �



would converge.

∞

∑n � 1

� � 1 � n � xn � � n � 10 �� 6 � nThe series is convergentfrom x = , left end included (Y,N):

to x = , right end included(Y,N):


∞

∑n � 1

� 9 � n � xn � � n � 1 �� n � 10 �The series is convergentfrom x = , left end included (Y,N):

to x = , right end included(Y,N):

9.(1 pt) Find the interval of convergence for the given powerseries.

∞

∑n � 1

n5 � x � 3 � n� 4n � � n 17

3 �The series is convergent:

from x = , left end included (Y,N):to x = , right end included (Y,N):

10.(1 pt) Match each of the power series with its interval ofconvergence.

1.∞

∑n � 1

� x � 2 � n� 2 � n2.

∞

∑n � 1

� 2x � nn2

3.∞

∑n � 1

� x � 2 � n� n! � 2n

4.∞

∑n � 1

n! � 2x � 2 � n2n

A. � � 12 � 1

2 �B.

�2 � 2 �

C. � � ∞ � ∞ �D. � 0 � 4 �

11.(1 pt) Suppose that9x� 10 � x � �

∞

∑n � 0

cnxn �Find the first few coefficients.c0 �

c1 �c2 �c3 �c4 �Find the radius of convergence R of the power series.

R � .

12.(1 pt) The function f � x � � 10�1 � 6x � 2 is represented as a

power series1

f � x � � ∞

∑n � 0

cnxn �Find the first few coefficients in the power series.c0 �

c1 �c2 �c3 �c4 �Find the radius of convergence R of the series.

R � .

13.(1 pt) The function f � x � � 10�1�

10x � 2 is represented as apower series

f � x � � ∞

∑n � 0

cnxn �Find the first few coefficients in the power series.c0 �

c1 �c2 �c3 �c4 �Find the radius of convergence R of the series.

R � .

14.(1 pt) The function f � x � � 61�

36x2 is represented as apower series

f � x � � ∞

∑n � 0

cnxn �Find the first few coefficients in the power series.

c0 �c1 �c2 �c3 �c4 �Find the radius of convergence R of the series.

R � .

15.(1 pt) The function f � x � � 4x2 arctan � x7 � is represented asa power series

f � x � � ∞

∑n � 0

cnxn �What is the lowest term with a nonzero coefficient.

Find the radius of convergence R of the series.R � .

16.(1 pt) The function f � x � � 10xarctan � 6x � is representedas a power series

f � x � � ∞

∑n � 0


c0 �c1 �c2 �c3 �c4 �

Find the radius of convergence R of the series.

R � .17.(1 pt) The function f � x � � ln � 4 � x � is represented as a

power series

f � x � � ∞

∑n � 0


c0 �c1 �c2 �c3 �c4 �


18.(1 pt) The function f � x � � 4x ln � 1 � x � is represented as apower series

f � x � � ∞

∑n � 0

cnxn �Find the FOLLOWING coefficients in the power series.

c2 �c3 �c4 �c5 �c6 �


19.(1 pt) The function f � x � � 7x ln � 1 � 2x � is represented asa power series

f � x � � ∞

∑n � 0

cnxn �Find the FOLLOWING coefficients in the power series.

c0 �c1 �c2 �c3 �c4 �


20.(1 pt) Represent the function 6�1 � 6x � as a power series

f � x � � ∞

∑n � 0

cnxn

c0 �c1 �c2 �c3 �c4 �Find the radius of convergence R � .

21.(1 pt) The function f � x � � ln � 1 � x2 � is represented as apower seriesf � x � � ∑∞

n � 0 cnxn �Find the FOLLOWING coefficients in the power series.

c0 �c1 �c2 �

2

c3 �c4 �


22.(1 pt) (a)Evaluate the integral� 2

0

24x2 � 4

dx.

Your answer should be in the form kπ, where k is an integer.What is the value of k?

Hint: d arctan�x �

dx � 1x2 � 1

k �(b)Now, lets evaluate the same integral using power series. First,

find the power series for the function f � x � � 24x2 � 4

. Then, inte-grate it from 0 to 2, and call it S. S should be an infinite series.What are the first few terms of S ?

a0 �a1 �

a2 �a3 �a4 �

(c) The answers to part (a) and (b) are equal (why?). Hence,if you divide your infinite series from (b) by k (the answer to(a)), you have found an estimate for the value of π in terms of aninfinite series. Approximate the value of π by the first 5 terms.

.(d)What is the upper bound for your error of your estimate if you

use the first 10 terms? (Use the alternating series estimation.).

23.(1 pt) Define the double factorial of n, denoted n!!, as fol-lows:

n!! ��

1 � 3 � 5 �� n � 2 �� n if n is odd

2 � 4 � 6 �� n � 2 �� n if n is even

where � � 1 � !! � 0!! � 1.Find the radius of convergence for the given power series.

∞

∑n � 1

2n � n! � � 4n � 5 � ! � � 2n � !!9n � � � n � 2 � ! � 4 � � 4n � 2 � !! � � 9x � 7 � n

The radius of convergence, R =24.(1 pt) POWER SERIES AND TAYLOR POLYNOMI-

ALSPower Series

A power series∞

∑n � 0

anxn has a RADIUS OF CONVER-

GENCE r.

The series converges for�x

� �r and diverges for

�x

�� r�

The radius of convergence is usually calculated by the ratio test,applied to the terms of the power series.

Suppose that limn � ∞

� an�

1

an

�exists. Then the power series con-

verges if�x

�limn � ∞

� an�

1

an

� � 1 and diverges if�x

�limn � ∞

� an�

1

an

�� 1 � The radius

of convergence is r ��

limn � ∞

� an�

1

an

� � � 1

�To determine whether the power series converges when x � r�replace x by r in the power series and decide whether the result-

ing numerical series,∞

∑n � 0

anrn converges. The ratio test will not

help in deciding this. Use some other convergence test.To determine whether the power series converges when x � � r,proceed analogously.

Taylor and MacLaurin series

If f � x � � ∞

∑n � 0

anxn converges in some interval � � s � s � containing

the point zero, then for each n :an � 1

n! f�n � � 0 � �

Power series may be integrated or differentiated term by term.That is:d fdx �

∞

∑n � 0� n � 1 � an

�1xn �

� x

0f � t � dt � ∞

∑n � 1

�1n � an � 1xn �

The nth degree MacLaurin polynomial for f(x) is

Tn � x � � n

∑j � 0

f�j � � 0 �j!

x j

It approximates f(x) with error Rn � x � .That is, f � x � � Tn � x � � Rn � x � � The size of the error is estimatedby

�Rn � x � � �

M � x � � n � 1 ��n�

1 � ! �Here, M is an upper bound for the (n+1)-st derivative of f be-tween 0 and x � It is enough that�f�n�

1 � � t � � � M for all t such that�t

� � �x

� �For every statement above you should know the analogous state-ment for a power series in powers of � x � c � which has the form

∞

∑n � 0

an � x � c � n �To receive a point enter the letter y.

answer


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Series9Taylor due 5/9/06 at 2:00 AM.

1.(1 pt) Compute the 9th derivative of

f � x � � arctan

�x3

6 �at x � 0.

f�9 � � 0 � �

Hint: Use the MacLaurin series for f � x � .2.(1 pt) Compute the 7th derivative of

f � x � � cos � 2x3 � � 1

x5

at x � 0.f�7 � � 0 � �

Hint: Use the MacLaurin series for f � x � .3.(1 pt) Find the degree 3 Taylor polynomial T3 � x � of func-

tionf � x � � � � 7x � 58 � 3 2

at a � 6.T3 � x � �

4.(1 pt) The Taylor series for f � x � � x3 at -2 is∞

∑n � 0

cn � x � 2 � n �Find the first few coefficients.c0 �

c1 �c2 �c3 �c4 �5.(1 pt) The Taylor series for f � x � � ln � sec � x � � at a � 0 is

∞

∑n � 0

cn � x � n �Find the first few coefficients.c0 �

c1 �c2 �c3 �c4 �

Find the exact error in approximating ln � sec � 0 � 3 � � by its fourthdegree Taylor polynomial at a � 0 �The error is

6.(1 pt) The Taylor series for f � x � � ln � sec � x � � at a � 0 is∞

∑n � 0

cn � x � n �Find the first few coefficients.c0 �

c1 �c2 �c3 �c4 �

7.(1 pt) Find T4 � x � : the Taylor polynomial of degree 4 of thefunction f � x � � arctan � 13x � at a � 0.

(You need to enter a function.)T4 � x � �

8.(1 pt) Find T4 � x � : the Taylor polynomial of degree 4 of thefunction f � x � � arctan � x1 � at a � 0.(You need to enter a function.)

T4 � x � �9.(1 pt) Let T5 � x � be the fifth degree Taylor polynomial of the

function f � x � � cos � 0 � 5x � at a � 0.A. Find T5 � x � � (Enter a function.)

T5 � x � �B. Find the largest integer k such that for all x for which

�x

� � 1the Taylor polynomial T5 � x � approximates f � x � with errorless than 1

10k �k �

10.(1 pt) Match each of the Maclaurin series with right func-tion.

1.∞

∑n � 0� � 1 � n x2n

�1

� 2n � 1 � !2.

∞

∑n � 0

� � 1 � nx2n

� 2n � !3.

∞

∑n � 0

� � 1 � nx2n�

1

2n � 1

4.∞

∑n � 0

xn

n!

A. arctan � x �B. ex

C. cos � x �D. sin � x �

11.(1 pt) Match each of the Maclaurin series with right func-tion.

1.∞

∑n � 0

� � 1 � n22nx2n

� 2n � !2.

∞

∑n � 0

� � 1 � n2x2n�

1

2n � 1

3.∞

∑n � 0� � 1 � n 2x2n

�1

� 2n � 1 � !4.

∞

∑n � 0

2nxn

n!

A. 2arctan � x �B. cos � 2x �C. e2x

D. 2sin � x �12.(1 pt) Select the FIRST correct reason why the given se-

ries diverges.

A. sin(x)1

B. exp(x)C. cos(x)D. arctan(x)

1.∞

∑n � 0� � 1 � n x2n

�1

� 2n � 1 � !2.

∞

∑n � 0

� � 1 � nx2n�

1

2n � 1

3.∞

∑n � 0

� � 1 � nx2n

� 2n � !4.

∞

∑n � 0

xn

n!


A. sin(2x)B. exp(2x)C. cos(2x)D. arctan(2x)

1.∞

∑n � 0� � 1 � n � 2x � 2n

�1

� 2n � 1 � !2.

∞

∑n � 0

2nxn

n!

3.∞

∑n � 0

� � 1 � n � 2x � 2n�

1

2n � 1

4.∞

∑n � 0

� � 1 � n22nx2n

� 2n � !14.(1 pt) Let F � x � � � x

0sin � 8t2 � dt.

Find the MacLaurin polynomial of degree 7 for F � x � .Use this polynomial to estimate the value of� 0 77

0sin � 8x2 � dx.

15.(1 pt) Let F � x � � � x

0e � 2t4

dt.

Find the MacLaurin polynomial of degree 5 for F � x � .Use this polynomial to estimate the value of

� 0 14

0e � 2x4

dx.

16.(1 pt) Find the Maclaurin series of the function f � x � �3x3 � 4x2 � 5x � 4

� f � x � � ∞

∑n � 0

cnxn �c0 �c1 �c2 �c3 �c4 �Find the radius of convergence R � Enter INF if the

radius of covergence is infinity .

17.(1 pt) Represent the function x0 2 as a power series∞

∑n � 0

cn � x � 8 � n �c0 �c1 �c2 �c3 �Find the left endpoint of the interval of convergence.left end = .Find the right endpoint of the interval of convergence.right end = .

18.(1 pt) Find Taylor series of function f � x � � ln � x � at a � 8.

� f � x � � ∞

∑n � 0

cn � x � 8 � n �c0 �c1 �c2 �c3 �c4 �Find the interval of convergence.The series is convergent:from x = , left end included (Y,N):

to x = , right end included (Y,N):

19.(1 pt) Evaluate

limx � 0

ln � 1 � x � � x � x2

2

11x3

Hint: Using power series.

20.(1 pt) Evaluate

limx � 0

e � 3x3 � 1 � 3x3 � 92 x6

15x9


21.(1 pt) Assume that sin � x � equals its Maclaurin series forall x.Use the Maclaurin series for sin � 7x2 � to evaluate the integral

� 0 63

0sin � 7x2 � dx

.Your answer will be an infinite series. Use the first two terms

to estimate its value.

22.(1 pt) Assume that ex equals its Maclaurin series for all x.Use the Maclaurin series for e � 2x4

to evaluate the integral

� 0 11

0e � 2x4

dx

Your answer will be an infinite series. Use the first two termsto estimate its value.

23.(1 pt) The Taylor series for f � x � � ex at a = 1 is∞

∑n � 0

cn � x �1 � n �

2

Find the first few coefficients.c0 �

c1 �c2 �c3 �c4 �24.(1 pt) The Taylor series for f � x � � sin � x � at a = π

2 is∞

∑n � 0

cn � x � π2� n �


c1 �c2 �c3 �c4 �25.(1 pt) The Taylor series for f � x � � sin � x � at a = π

3 is∞

∑n � 0

cn � x � π3� n �


c1 �c2 �c3 �c4 �26.(1 pt) The Taylor series for f � x � � cos � x � at a = π

2 is∞

∑n � 0

cn � x � π2� n �


c1 �c2 �c3 �c4 �27.(1 pt) The Taylor series for f � x � � cos � x � at a = π

4 is∞

∑n � 0

cn � x � π4� n �


c1 �c2 �c3 �c4 �28.(1 pt) Find the Maclaurin series of the function f � x � �� 5x � arctan � 7x2 �

� f � x � � ∞

∑n � 0

cnxn �c3 �c5 �c7 �c9 �c11 �

29.(1 pt) Find the Maclaurin series of the function f � x � �� 2x2 � e � 6x

� f � x � � ∞

∑n � 0

cnxn �c1 �c2 �c3 �c4 �c5 �30.(1 pt) Find the Maclaurin series of the function f � x � �� 3x2 � sin � 9x �

� f � x � � ∞

∑n � 0

cnxn �c3 �c4 �c5 �c6 �c7 �31.(1 pt) Find the Maclaurin series of the function f � x � �

2cos � 7x2 �� f � x � � ∞

∑n � 0

cnxn �c0 �c2 �c4 �c6 �c8 �32.(1 pt) Match each of the Maclaurin series with right func-

tion.

1.∞

∑n � 0

� � 1 � n32n�

1

2n � 1

2.∞

∑n � 0� � 1 � n 32n

�1

� 2n � 1 � !3.

∞

∑n � 0

3n

n!

4.∞

∑n � 0

� � 1 � n32n

� 2n � !A. e3

B. cos � 3 �C. sin � 3 �D. arctan � 3 �

33.(1 pt) Find T5 � x � : Taylor polynomial of degree 5 of thefunction f � x � � cos � x � at a � 0.

(You need to enter function.)T5 � x � �Find all values of x for which this approximation is within0.004234 of the right answer. Assume for simplicity that welimit ourselves to

�x

� �1.�

x� �34.(1 pt) Let T4 � x � : be the Taylor polynomial of degree 4 of

the function f � x � � cos � x � at a � 0.3

Suppose you approximate f � x � by T4 � x � , and if�x

� �1, what

is the bound for your error of your estimate? (Hint: use thealternating series approximation.)

35.(1 pt) Let Tk � x � : be the Taylor polynomial of degree k ofthe function f � x � � sin � x � at a � 0.

Suppose you approximate f � x � by Tk � x � , and if�x

� �1, how

many terms do you need (that is, what is k) for you to have yourerror to be less than 1

120 ? (Hint: use the alternating series ap-proximation.)

36.(1 pt) Let T8 � x � : be the Taylor polynomial of degree 8 ofthe function f � x � � ln � 1 � x � at a � 0.

Suppose you approximate f � x � by T8 � x � , find all positive val-ues of x for which this approximation is within 0.001 of the rightanswer. (Hint: use the alternating series approximation.)

0 �x

�37.(1 pt) Represent the function 3 ln � 10 � x � as a power series

(Maclaurin series) � f � x � � ∞

∑n � 0

cnxn �c0 �c1 �c2 �c3 �c4 �Find the radius of convergence R � .

38.(1 pt) Evaluate

limx � 0

cos � x � � 1 � x2

2

8x4



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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vectors0Introduction due 1/3/08 at 2:00 AM.

1.(1 pt) If Tom Bombadil’s house is 24 miles east of Hob-biton and 32 miles south, what is the straight line distance (omitunits)?

2.(1 pt) If the distance from the town of Bree to Weathertopis 9 miles on a 45 degree upward slope, what is the elevationgain (omit units)?

3.(1 pt) Frodo and Sam are studying a topographic map ofMordor. Place the letter describing contour lines on a map tothe left of the number describing a possible goal.

1. If Frodo and Sam want to find a level route, they shouldlook at:

2. If Frodo and Sam want to go directly uphill, they shouldgo:

3. If Frodo and Sam want to find the River Anduin, theyshould look for:

4. If Frodo and Sam want to find Mount Doom, theyshould look for:

A. Single contour linesB. Perpendicular to the contour linesC. Concentric contour linesD. Parallel contour lines

4.(1 pt) The nine Ring Wraiths want to fly from Barad-Dur toRivendell. Rivendell is directly north of Barad-Dur. The DarkTower reports that the wind is coming from the west at 61 milesper hour. In order to travel in a straight line, the Ring Wraithsdecide to head northwest. At what speed should they fly (omitunits)?

5.(1 pt) As Aragorn views the Dark Lord in a crystal ball ofradius 2, he realizes that:

The surface area of the ball equals:

The volume of the ball equals:

6.(1 pt) As Gandalf falls into the depths of Moria, he beginsto spin. If he wishes to slow his rate of spinning, he should dowhich of the following (type the appropriate letter)?

A. Spread his arms wideB. Think of SauronC. Wiggle His NoseD. Hug HimselfE. Think of Galadriel

7.(1 pt) The population of Elves in Lorien is constant. If fiveElves per day cross outward over the boundary of Lorien, andnone ever return, then we can conclude that Elves in Lorien arebeing:

A. PersecutedB. Attracted by the promise of a better lifeC. BornD. Anti-social

8.(1 pt) In the land of Mordor, where the shadows lie, it isafternoon.

In which direction do the shadows point? You may assumethat the Earth’s axis of rotation is perpendicular to its plane ofrevolution about the sun. Type N, S, W, or E as appropriate.

Type C, D, or I depending on whether the lengths of the shad-ows are constant, decreasing, or increasing.

9.(1 pt) Two dwarves decide to bore a tunnel through the cen-ter of the earth, connecting the mines of Moria with its antipode.They each have identical drills. One dwarf begins drilling fromMoria and the other dwarf begins drilling from the antipode.When they meet at the center of the earth, are their two drillsturning in the same direction? Type Y if yes, N if no.


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vectors1space3D due 1/4/08 at 2:00 AM.

1.(1 pt)What is the distance from the point (10, 7, -2) to the xz-

plane?Distance =

2.(1 pt) What do the following equations represent in R3?Match the two sets of letters:

a. a vertical planeb. a horizontal planec. a plane which is neither vertical nor horizontal

A. � 10x � 7y � 8B. x � � 9C. y � � 3D. z � � 3

3.(1 pt)Find an equation of the sphere with center (5, 5, -2) and ra-

dius 5.= 0

Note that you must move everything to the left hand side ofthe equation and that we desire the coefficients of the quadraticterms to be 1.

4.(1 pt)Find an equation of the sphere that passes through the origin

and whose center is (6, -7, -4).

= 0Note that you must put everything on the left hand side of theequation and that we desire the coefficients of the quadraticterms to be 1.

5.(1 pt)Find the center and radius of the spherex2 � 16x � y2 � 4y � z2 � 4z � � 68Center: ( , , )Radius:

6.(1 pt)Find the equation of a sphere if one of its diameters has end-

points: (-9, -7, -1) and (9, 11, 17).= 0

Note that you must move everything to the left hand side ofthe equation and that we desire the coefficients of the quadraticterms to be 1.

7.(1 pt)Find an equation of the largest sphere with center (1, 7 , 8)

that is contained completely in the first octant.= 0

Note that you must move everything to the left hand side of theequation that we desire the coefficients of the quadratic terms tobe 1.


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vectors2DotProduct due 1/5/08 at 2:00 AM.

1.(1 pt)Let a = (4, -3, -6) and b = (-6, -6, -9) be vectors. Compute

the following vectors.A. a + b = ( , , )B. -6a= ( , , )C. a - b= ( , , )D.

�a

�=

2.(1 pt)A child walks due east on the deck of a ship at 2 miles per

hour.The ship is moving north at a speed of 19 miles per hour.

Find the speed and direction of the child relative to the sur-face of the water.

Speed = mphThe angle of the direction from the north = (radians)

3.(1 pt)A horizontal clothesline is tied between 2 poles, 10 meters

apart.When a mass of 5 kilograms is tied to the middle of the clothes-line, it sags a distance of 2 meters.

What is the magnitude of the tension on the ends of theclothesline?

Tension = N4.(1 pt)Find a � b if�

a�= 4,�

b�= 5,

and the angle between a and b is π� 8 radians.a � b =5.(1 pt)If a = (3, -7, 0) and b = (8, -1, 1),find a � b = .6.(1 pt)What is the angle in radians between the vectors

a = (-2, -8, -10) andb = (4, -9, 6)?

Angle: (radians)

7.(1 pt)Find a unit vector in the same direction as a = (-5, -5, 5).( ,

,)

8.(1 pt)Let a = (1, 9, 3) and b = (8, -9, -10) be vectors. Find the

scalar, vector, and orthogonal projections of b onto a.Scalar Projection:Vector Projection:

( ,,)

Orthogonal Projection:( ,

,)

9.(1 pt)A constant force F � � 10i � 6j � 7k moves an object along a

straight line from point (5, -10, 0) to point (0, -5, 4).Find the work done if the distance is measured in meters and

the magnitude of the force is measured in newtons.Work: Nm10.(1 pt)A woman exerts a horizontal force of 1 pounds on a box as

she pushes it up a ramp that is 9 feet long and inclined at anangle of 30 degrees above the horizontal.

Find the work done on the box.Work: ft-lb11.(1 pt)Gandalf the Grey started in the Forest of Mirkwood at a point

with coordinates (-2, -1) and arrived in the Iron Hills at the pointwith coordinates (0, 4). If he began walking in the direction ofthe vector v � 4i � 2j and changes direction only once, whenhe turns at a right angle, what are the coordinates of the pointwhere he makes the turn.

( , )12.(1 pt)If Yoda says to Luke Skywalker, “The Force be with you,”

then the dot product of the Force and Luke should be:� A. zero� B. negative� C. any real number� D. positive


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vectors3CrossProduct due 1/6/08 at 2:00 AM.

1.(1 pt) You are looking down at a map. A vector u with�u

�

= 6 points north and a vector v with�v

�= 8 points northeast. The

crossproduct u � v points:A) south

B) northwestC) upD) down

Please enter the letter of the correct answer:The magnitude

�u � v

�=

2.(1 pt) Let a = (7, 8, 3) and b = (10, 7, 5) be vectors.

Compute the cross product a � b. ( , , )

3.(1 pt)If a � i � j � 5k and b � i � j � 5k

Compute the cross product a � b.i + j + k

4.(1 pt) If a � i � 4j � k and b � i � 8j � k, find a unit vectorwith positive first coordinate orthogonal to both a and b.

i + j + k5.(1 pt) Find the area of the parallelogram with vertices (5,4),

(9, 8), (12, 11), and (16, 15).


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vectors4PlanesLines due 1/7/08 at 2:00 AM.

1.(1 pt) Find a unit vector with positive first coordinate thatis orthogonal to the plane through the points P = (-2, 5, 5), Q =(3, 10, 10), and R = (3, 10, 14).( , , )

2.(1 pt) A bicycle pedal is pushed straight downwards by afoot with a 48 Newton force. The shaft of the pedal is 20 cmlong. If the shaft is π � 5 radians past horizontal, what is themagnitude of the torque about the point where the shaft is at-tached to the bicycle?Nm

3.(1 pt) Enter T or F depending on whether the statement istrue or false. (You must enter T or F – True and False will notwork.)

1. Two lines parallel to a plane are parallel.2. Two planes parallel to a third plane are parallel.3. Two lines perpendicular to a plane are parallel4. Two planes parallel to a line are parallel.5. Two planes perpendicular to a third plane are parallel.6. A plane and a line either intersect or are parallel.7. Two lines perpendicular to a third line are parallel.8. Two planes perpendicular to a line are parallel.9. Two lines parallel to a third line are parallel.

10. Two lines either intersect or are parallel.11. Two planes either intersect or are parallel.

4.(1 pt) Find a vector equation for the line through the pointP = (-5, -2, 5) and parallel to the vector v = (4, 3, -5).Assume r � 0 � � � 5i � 2j � 5k and that v is the velocity vector ofthe line.r(t) = i + j + k

5.(1 pt) Find a vector equation for the line through the pointP = (-3, 3, 4) and parallel to the vector v = (-2, -5, -5).Assume r � 0 � � � 3i � 3j � 4k and that v is the velocity vector ofthe line..r � t � = i + j + k

Rewrite this in terms of the parametric equations for the line.x =

y =z =

6.(1 pt) Given a the vector equation r(t) = (-1 + -2t)i + (5 + -5t)j + (5 + -4t)k, rewrite this in terms of the parametric equationsfor the line.x(t) =y(t) =z(t) =

7.(1 pt) Given a the vector equation r(t) = (-5 + 4t)i + (-2 +1t)j + (0 + 3t)k, rewrite this in terms of the symmetric equationsfor the line.

(quotient involving x)(quotient involving y) =

(quotient involving z) =

8.(1 pt) Consider the planes 2x � 3y � 3z � 1 and 2x � 3z � 0 �(A) Find the unique point P on the y-axis which is on both

planes. ( , , )(B) Find a unit vector u with positive first coordinate that is

parallel to both planes.i + j + k

(C) Use parts (A) and (B) to find a vector equation for theline of intersection of the two planes,r � t � �

i + j + k9.(1 pt) (A) Find the parametric equations for the line through

the point P = (4, 1, 3) that is perpendicular to the plane5x � 4y � 4z � 1 �Use ”t” as your variable, t = 0 should correspond to P, and thevelocity vector of the line should be the same as the standardnormal vector of the plane.

x =y =z =

(B) At what point Q does this line intersect the yz-plane?Q = ( , , )

10.(1 pt) Consider the two linesL1 : x � � 2t � y � 1 � 2t � z � 3t andL2 : x � � 8 � 4s � y � 4 � 1s � z � 4 � 2s

Find the point of intersection of the two lines.P = ( , , )

11.(1 pt) Find an equation of the plane through the point (2, -1, -5) and perpendicular to the vector (-3, 0, 3). Do this problemin the standard way or WebWork may not recognize a correctanswer.

x + y + z =

12.(1 pt) Find an equation of the plane through the point (0,-4, 4) and parallel to the plane 3x � 1y � 2z � 3. Do this problemin the standard way or WebWork may not recognize a correctanswer.

x + y + z =

13.(1 pt) Find the point P where the line x = 1 + t, y = 2t, z =-3t intersects the plane x + y - z = 5.

P = ( , , )

14.(1 pt) Find the angle in radians between the planes 5x �z � 1 and � 2y � z � 1 �

15.(1 pt) Find the distance from the point (5, 3, 5) to the linex � 0 � y � 3 � 3t � z � 5 � 3t �

16.(1 pt) Find the distance from the point (1, 3, -3) to theplane � 2x � 4y � 1z � � 3 �

17.(1 pt) Match the surfaces with the appropriate descrip-tions.

1

1. x2 � 2y2 � 3z2 � 12. z � x2

3. z � 2x2 � 3y2

4. z � 2x � 3y5. z � y2 � 2x2

6. x2 � y2 � 57. z � 4

A. horizontal planeB. ellipsoidC. nonhorizontal plane

D. parabolic cylinderE. circular cylinderF. hyperbolic paraboloid

G. elliptic paraboloid

18.(1 pt) A million years ago, an alien species built a verticaltower on a horizontal plane. When they returned they discov-ered that the ground had tilted so that measurements of 3 pointson the ground gave coordinates of (0, 0, 0), (2, 1, 0), and (0, 1,2). By what angle does the tower now deviate from the vertical?

radians.


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vectors5Coordinates due 1/8/08 at 2:00 AM.

1.(1 pt) What are the rectangular coordinates of the pointwhose cylindrical coordinatesare � r � 0 � θ � 1 � 6 � z � 2 � ?x =y =z =

2.(1 pt)What are the rectangular coordinates of the point whose

cylindrical coordinates are� r � 7 � θ � 4π7 � z � � 8 � ?

x =y =z =

3.(1 pt)What are the cylindrical coordinates of the point whose rect-

angular coordinates are � x � 1 � y � 2 � z � 4 � ?r =θ =z =

4.(1 pt)What are the cylindrical coordinates of the point whose rect-

angular coordinates are � x � � 4 � y � 2 � z � � 5 � ?r =θ =z =

5.(1 pt)What are the rectangular coordinates of the point whose

spherical coordinates are� 4 � � 3π6 � � 1π

6� ?

x =y =z =

6.(1 pt)What are the spherical coordinates of the point whose rect-

angular coordinates are� 5 � 2 � 2 � ?

ρ =θ =φ =

7.(1 pt)What are the cylindrical coordinates of the point whose

spherical coordinates are� 4 � 5 � 5π6� ?

r =θ =z=

8.(1 pt)Match the given equation with the verbal description of the

surface:

A. PlaneB. Elliptic or Circular ParaboloidC. ConeD. Half planeE. Circular CylinderF. Sphere

1. r � 42. ρcos � φ � � 43. ρ � 44. φ � π

35. z � r2

6. ρ � 2cos � φ �7. r � 2cos � θ �8. r2 � z2 � 169. θ � π

3

9.(1 pt)If an astronomer is using polar coordinates, then which of the

following is the most likely object of study?� A. a planet� B. a globular cluster� C. a solar system� D. the whole universe


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VecFunction1Curves due 2/3/08 at 2:00 AM.

1.(1 pt)If r � t � � � �

t � 2 � i � t2 � 0t � 0 j � sin � � 5πt � k.

Thenlimt � 1

r � t � = i � j � k.

2.(1 pt)The curve (x=cost, y=sint, z=t) lies on which of the following

surfaces.Enter T or F depending on whether the statement is true or false.(You must enter T or F – True and False will not work.)

1. a sphere2. an elliptic paraboloid3. a circular cylinder4. a plane

3.(1 pt)For the given position vectors r � t � compute the tangent ve-

locity vector r� � t � for the given value of t .

A.)If r � t � � � cos5t � sin5t �Then r

� � π4� = ( , )

B.)If r � t � � � t2 � t3 �Then r

� � 2 � = ( , )C.)If r � t � � e5t i � e � 2tj � tk.

Then r� � � 5 � = i � j � k .

4.(1 pt)For the given position vectors r � t � compute the unit tangent

vector T � t � for the given value of t .A.)

If r � t � � � cos2t � sin2t �Then T � π

4� = ( , )

C.)If r � t � � � t2 � t3 �Then T � 2 � = ( , )

C.)If r � t � � e2t i � e � 2tj � tk.Then T � � 1 � = i � j � k .

5.(1 pt)Find parametric equations for the tangent line at the point� cos � 3π

6� � sin � 3π

6� � 3π

6�� on the curve x � cost � y � sin t � z � t

x � t � =y � t � =z � t � =

6.(1 pt)Evalute� 3

0 � ti � t2j � t3k � dt = i � j � k.

7.(1 pt) If r � t � � cos � 4t � i � sin � 4t � j � 4tkcompute r

� � t � = i � j � kand

�r � t � dt= i � j � k

8.(1 pt)A particle in space undergoes a constant nonzero accelera-

tion. Depending on the circumstances, the particle’s trajectorycan be held by the following curves.Enter T or F depending on whether the statement is true or false.(You must enter T or F – True and False will not work.)

1. a hyperbola2. an ellipse3. a strait line4. a parabola5. a circle


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VecFunction2Curvature due 2/4/08 at 2:00 AM.

1.(1 pt) Find the length of the given curve:

r � t � � � � 2t � � 1sint �� 1cost �where � 2

�t

�1.

2.(1 pt) Starting from the point � 3 � 4 �� 1 � reparametrize thecurve

r � t � � � 3 � 3t � i � � 4 � 3t � j � � � 1 � 3t � kin terms of arclength.

r � t � s � � � i � j � k

3.(1 pt) If r � t � � cos � � 4t � i � sin � � 4t � j � 10tk, compute:A. The velocity vector v � t � � i � j � kB. The acceleration vector a � t � � i � j � k

Note: the coefficients in your answers must be entered in theform of expressions in the variable t; e.g. “5 cos(2t)”

4.(1 pt) Consider the helix r � t � � � cos � 2t � � sin � 2t � �� 4t � .Compute, at t � π

6 :A. The unit tangent vector T � ( , , )B. The unit normal vector N � ( , , )C. The unit binormal vector B � ( , , )D. The curvature κ �5.(1 pt) Find the curvature κ � t � of the curve r � t � � � 4sin t � i �� 4sin t � j � � 1cost � k6.(1 pt) Find the curvature of y � sin � 5x � at x � π

4 .


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VecFunction3Motion due 2/5/08 at 2:00 AM.

1.(1 pt) Given that the acceleration vector is a � t � �� 9cos � � 3t � � i � � � 9sin � � 3t � � j � � 4t � k, the initial velocity isv � 0 � � i � k, and the initial position vector is r � 0 � � i � j � k,compute:

A. The velocity vector v � t � � i � j � kB. The position vector r � t � � i � j � kNote: the coefficients in your answers must be entered in the

form of expressions in the variable t; e.g. “5 cos(2t)”

2.(1 pt) A gun has a muzzle speed of 70 meters per second.What angle of elevation should be used to hit an object 150 me-ters away? Neglect air resistance and use g � 9 � 8m � sec2 as theacceleration of gravity.

radians

3.(1 pt) Match the parametric equations with the verbal de-scriptions of the surfaces by putting the letter of the verbal de-scription to the left of the letter of the parametric equation.

1. r � u � v � � ui � cosvj � sinvk2. r � u � v � � ui � vj � � 2u � 3v � k3. r � u � v � � ucosvi � usinvj � u2k4. r � u � v � � ui � ucosvj � usinvkA. coneB. circular paraboloidC. planeD. circular cylinder

4.(1 pt) A factory has a machine which bends wire at a rateof 10 unit(s) of curvature per second. How long does it take tobend a straight wire into a circle of radius 9?

seconds


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable1Functions due 3/3/08 at 2:00 AM.

1.(1 pt)Match the surfaces with the verbal description of the level

curves by placing the letter of the verbal description to the leftof the number of the surface.

1. z � � � x2 � y2 �2. z � x2 � y2

3. z � � � 25 � x2 � y2 �4. z � 2x2 � 3y2

5. z � 2x � 3y6. z � 1

x � 17. z � xy

A. a collection of equally spaced parallel linesB. a collection of concentric ellipsesC. a collection of unequally spaced concentric circlesD. a collection of unequally spaced parallel linesE. a collection of equally spaced concentric circlesF. two straight lines and a collection of hyperbolas

2.(1 pt)Match the functions with the verbal description of the level

surfaces by placing the letter of the verbal description to the leftof the number of the function.

1. w � x2 � 2y2 � 3z2

2. w � � � x2 � 2y2 � 3z2 �3. w � � � x2 � y2 � z2 �4. w � x � 2y � 3z5. w � x2 � y2 � z2

6. w � � � x � 2y � 3z �7. w � x2 � y2 � z2

A. a collection of unequally spaced parallel planesB. two cones and two collections of hyperboloidsC. a collection of equally spaced concentric spheresD. a collection of unequally spaced concentric spheres

E. a collection of equally spaced parallel planesF. a collection of concentric ellipsoids

3.(1 pt)Each of the following functions has a set on which it is con-

tinuous and that set has a boundary. Match the verbal descrip-tion of this boundary with the function by putting the letter ofthe boundary to the left of the letter of the function.

1. f � x � y � z � � 1x2 � y2 � z2

2. f � x � y � z � � xyzx2 � y2 � z

3. f � x � y � � x lny4. f � x � y � � 1

4 � x2 � y2

5. f � x � y � z � � z1 � x2 � y2

A. a straight lineB. a circular parabaloidC. a circular cylinderD. one pointE. a circle

4.(1 pt)The level curves of a function f � x � y � consist of a collection

of hyperbolas and two lines. If the lines intersect at a point P,what are the possibilities for P? Type the letters of all possibili-ties, with no punctuation, in alphabetical order.

A. P is a local maximum, that is, f � P � � f � Q � for all Q nearP.

B. P is a local minimum, that is, f � P � �f � Q � for all Q near

P.C. P is neither a local maximum nor a local minimum.

5.(1 pt)On a map showing the grave of George Mallory, the contour

lines are:� A. closely spaced� B. far apart


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable2Limits due 3/4/08 at 2:00 AM.

1.(1 pt)Find the limit, if it exists, or type N if it does not exist.

lim�x � y � � � 0 � � 4 � e

�5x2 � 3y2 �


lim�x � y � � � 0 � 0 � 5x2

1x2 � 4y2 �3.(1 pt)Find the limit, if it exists, or type N if it does not exist.

lim�x � y � � � 0 � 0 � � x � 8y � 2

x2 � 82y2 �


(Hint: use polar coordinates.)

lim�x � y � � � 0 � 0 � 7x3 � 4y3

x2 � y2 �5.(1 pt)Find the limit, if it exists, or type N if it does not exist.

lim�x � y � z � � � 3 � 5 � 4 � 1zex2 � y2

3x2 � 5y2 � 4z2 �6.(1 pt)Find the limit, if it exists, or type N if it does not exist.

lim�x � y � z � � � 0 � 0 � 0 � 1xy � 5yz � 3xz

1x2 � 25y2 � 9z2 �


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable3ParDer due 3/5/08 at 2:00 AM.

1.(1 pt)Find the first partial derivatives of f � x � y � � 2x � 3y

2x�

3y at the point(x,y) = (1, 2).

∂ f∂x � 1 � 2 � �∂ f∂y � 1 � 2 � �2.(1 pt)Find the first partial derivatives of f � x � y � z � � z arctan � y

x� at

the point (4, 4, -5).A. ∂ f

∂x � 4 � 4 � � 5 � �B. ∂ f

∂y � 4 � 4 � � 5 � �C. ∂ f

∂z � 4 � 4 � � 5 � �3.(1 pt)Find the first partial derivatives of f � x � y � � sin � x � y � at the

point (1, 1).A. fx � 1 � 1 � �B. fy � 1 � 1 � �

4.(1 pt)If sin � � 4x � 1y � z � � 0, find the first partial derivatives ∂z

∂x

and ∂z∂y at the point (0, 0, 0).

A. ∂z∂x � 0 � 0 � 0 � �

B. ∂z∂y � 0 � 0 � 0 � �

5.(1 pt)Find all the first and second order partial derivatives of

f � x � y � � � 2sin � 2x � y � � 7cos � x � y � .A. ∂ f

∂x � fx �B. ∂ f

∂y � fy �C. ∂2 f

∂x2 � fxx �D. ∂2 f

∂y2 � fyy �E. ∂2 f

∂x∂y � fyx �F. ∂2 f

∂y∂x � fxy �


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable4Linearization due 3/5/08 at 2:00 AM.

1.(1 pt) Find the equation of the tangent plane to the surfacez � 1y2 � 16x2 at the point � 4 � 3 �� 247 � .

z =Note: Your answer should be an expression of x and y; e.g.

“3x - 4y + 6”2.(1 pt) Find the equation of the tangent plane to the surface

z � e � 2x 17 ln � 1y � at the point � � 3 � 2 � 0 � 9865 � .z =Note: Your answer should be an expression of x and y; e.g.

“5x + 2y - 3”

3.(1 pt) Find the linearization L � x � y � of the functionf � x � y � � �

145 � 16x2 � 4y2 at � � 3 � 0 � .L � x � y � �Note: Your answer should be an expression in x and y; e.g.

“3x - 5y + 9”

4.(1 pt) Find the differential of the function w � xsin � 2yz1 � .dw = dx + dy + dzNote: Your answers should be expressions of x, y and z; e.g.

“3xy + 4z”

5.(1 pt) The dimensions of a closed rectangular box are mea-sured as 80 centimeters, 50 centimeters, and 50 centimeters, re-spectively, with the error in each measurement at most .2 cen-timeters. Use differentials to estimate the maximum error incalculating the surface area of the box.

square centimeters

6.(1 pt) Find an equation of the tangent plane to the para-metric surface x � 1r cosθ, y � 1r sinθ, z � r at the point�

1�

2 � 1�

2 � 2 � when r � 2, θ � π � 4.z =Note: Your answer should be an expression of x and y; e.g.

“3x - 4y”


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable5ChainRule due 3/6/08 at 2:00 AM.

1.(1 pt) Suppose w � xy � y

z , x � e4t , y � 2 � sin � 1t � , z �2 � cos � 5t � .

A. Use the chain rule to find dwdt as a function of x, y, z, and

t. Do not rewrite x, y, and z in terms of t, and do not rewrite e4t

as x.dwdt =

Note: Use exp() for the exponential function. Your answershould be an expression in x, y, z, and t; e.g. “3x - 4y”

B. Use part A to evaluate dwdt when t � 0.

2.(1 pt) Suppose z � x2 siny, x � � 2s2 � 0t2, y � � 4st.A. Use the chain rule to find ∂z

∂s and ∂z∂t as functions of x, y, s

and t.∂z∂s �∂z∂t �

B. Find the numerical values of ∂z∂s and ∂z

∂t when � s � t � �� 1 � � 3 � .

∂z∂s � � 1 �� 3 � �∂z∂t � � 1 �� 3 � �

3.(1 pt) The radius of a right circular cone is increasing ata rate of 4 inches per second and its height is decreasing at arate of 3 inches per second. At what rate is the volume of thecone changing when the radius is 20 inches and the height is 50inches?

cubic inches per second

4.(1 pt) In a simple electric circuit, Ohm’s law states thatV � IR, where V is the voltage in volts, I is the current in am-peres, and R is the resistance in ohms. Assume that, as the bat-tery wears out, the voltage decreases at 0.03 volts per secondand, as the resistor heats up, the resistance is increasing at 0.01ohms per second. When the resistance is 400 ohms and the cur-rent is 0.01 amperes, at what rate is the current changing?

amperes per second


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable6Gradient due 3/7/08 at 2:00 AM.

1.(1 pt) If f � x � y � � 2x2 � 2y2, find the value of the direc-tional derivative at the point � 1 � 2 � in the direction given by theangle θ � 2π

2 .

2.(1 pt) Suppose f � x � y � � � 3x2 � 3xy � 1y2, P � � 3 �� 2 � ,and u � � � 8

10 � 610

� .A. Compute the gradient of f.

∇ f � i � jNote: Your answers should be expressions of x and y; e.g. “3x -4y”

B. Evaluate the gradient at the point P.� ∇ f � � 3 � � 2 � � i � jNote: Your answers should be numbers

C. Compute the directional derivative of f at P in the directionu .� Du f � � P � �Note: Your answer should be a number

3.(1 pt) Suppose f � x � y � � xy , P �� 3 �� 1 � and v � 3i � 2j.

A. Find the gradient of f.∇ f � i � jNote: Your answers should be expressions of x and y; e.g. “3x -4y”

B. Find the gradient of f at the point P.� ∇ f � � P � � i � jNote: Your answers should be numbers

C. Find the directional derivative of f at P in the direction ofv.Du f �Note: Your answer should be a number

D. Find the maximum rate of change of f at P.

Note: Your answer should be a numberE. Find the (unit) direction vector in which the maximum rate

of change occurs at P.u � i � jNote: Your answers should be numbers

4.(1 pt) Suppose f � x � y � z � � xy � y

z , P �� 2 � 2 � 3 � .A. Find the gradient of f.

∇ f � i � j � kNote: Your answers should be expressions of x, y and z; e.g.“3x - 4y”

B. What is the maximum rate of change of f at the point P?

Note: Your answer should be a number5.(1 pt) Suppose that distances are measured in lightyears and

that the temperature T of a gaseous nebula is inversely propor-tional to the distance from a fixed point, which we take to be theorigin. Suppose that the temperature 1 lightyear from the originis 800 degrees celsius. Find the gradient of T at � x � y � z � .∇ f � i � j � kNote: Your answers should be expressions of x, y and z; e.g.“3x - 4y”

6.(1 pt) Consider the surface

4x2 � 1y2 � 4z2 � 9

and the point P �� 1 � 1 � 1 � on this surface.A. Starting with the equation x � 1 � 8t, find equations for

y and z which combine with this equation to give parametricequations for the normal line through P.y �z �Note: Your answers should be expressions of t; e.g. “3x - 4y”

B. Find an equation for the tangent plane through P.z �Note: Your answers should be expressions of x and y; e.g. ”3xy+ 2y”

7.(1 pt) The axis of a light in a lighthouse is tilted. When thelight points east, it is inclined upward at 10 degree(s). Whenit points north, it is inclined upward at 4 degree(s). What is itsmaximum angle of elevation?

degrees

8.(1 pt) You are hiking the Inca Trail on the way to MachuPicchu. When you arrive at the highest point on the trail, whichof the following are possibilities? In alphabetical order withoutpunctuation or spacing, list the letters which indicate possibili-ties.

(A) The path passes through the center of a set of concentriccontour lines.(B) The path is tangent to a contour line.(C) The path follows a contour line.(D) The path crosses a contour line.

possibilities:


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment Vmultivariable7MaxMin due 3/8/08 at 2:00 AM.

1.(1 pt) Suppose f � x � y � � x2 � y2 � 6x � 2y � 1(A) How many critical points does f have in R2?

(B) If there is a local minimum, what is the value of the dis-criminant D at that point? If there is none, type N.

(C) If there is a local maximum, what is the value of the dis-criminant D at that point? If there is none, type N.

(D) If there is a saddle point, what is the value of the discrim-inant D at that point? If there is none, type N.

(E) What is the maximum value of f on R2? If there is none,type N.

(F) What is the minimum value of f on R2? If there is none,type N.

2.(1 pt) Suppose f � x � y � � xy � ax � by.(A) How many local minimum points does f have in R2?

(The answer is an integer).

(B) How many local maximum points does f have in R2?

(C) How many saddle points does f have in R2?

3.(1 pt) Consider the function f � x � y � � xsin � y � .In the following questions, enter an integer value or type INFfor infinity.

(A) How many local minima does f have in R2?

(B) How many local maxima does f have in R2?

(C) How many saddle points does f have in R2?

4.(1 pt) Suppose f � x � y � � xy � 1 � 10x � 3y � .f � x � y � has 4 critical points. List them in increasing lexographicorder. By that we mean that (x, y) comes before (z, w) if x

�z

or if x � z and y�

w. Also, describe the type of critical point bytyping MA if it is a local maximum, MI if it is a local minimim,and S if it is a saddle point.

First point ( , ) of typeSecond point ( , ) of typeThird point ( , ) of typeFourth point ( , ) of type

5.(1 pt) Each of the following functions has at most one crit-ical point. Graph a few level curves and a few gradiants and, onthis basis alone, decide whether the critical point is a local max-imum (MA), a local minimum (MI), or a saddle point (S). Enter

the appropriate abbreviation for each question, or N if there isno critical point.

(A) f � x � y � � e � 2x2 � 4y2

Type of critical point:(B) f � x � y � � e2x2 � 4y2

Type of critical point:(C) f � x � y � � 2x2 � 4y2 � 4

Type of critical point:(D) f � x � y � � 2x � 4y � 4

Type of critical point:6.(1 pt) You are to manufacture a rectangular box with 3 di-

mensions x, y and z, and volume v � 125. Find the dimensionswhich minimize the surface area of this box.

x =y =z =

7.(1 pt) Find the coordinates of the point (x, y, z) on the planez = 1 x + 4 y + 3 which is closest to the origin.x =y =z =

8.(1 pt) Find the maximum and minimum values of f � x � y � �9x2 � 10y2 on the disk D: x2 � y2 �

1.maximum value:minimum value:

9.(1 pt) Find the maximum and minimum values of f � x � y � �4x � y on the ellipse x2 � 49y2 � 1maximum value:minimum value:

10.(1 pt) For each of the following functions, find the maxi-mum and mimimum values of the function on the circular disk:x2 � y2 �

1. Do this by looking at the level curves and gradiants.(A) f � x � y � � x � y � 4:

maximum value =minimum value =

(B) f � x � y � � 4x2 � 5y2:maximum value =minimum value =

(C) f � x � y � � 4x2 � 5y2:maximum value =minimum value =

11.(1 pt) For each of the following functions, find the max-imum and minimum values of the function on the rectanglarregion: � 4

�x

�4 � � 5

�y

�5.

Do this by looking at level curves and gradiants.(A) f � x � y � � x � y � 3:

maximum value =minimum value =

(B) f � x � y � � 3x2 � 4y2:maximum value =minimum value =

1

(C) = f � x � y � �� 5 � 2x2 � � 4 � 2y2:maximum value =minimum value =

12.(1 pt) Find the maximum and minimum values off � x � y � z � � 5x � 3y � 1z on the sphere x2 � y2 � z2 � 1.maximum value =minimum value =

13.(1 pt) Find the maximum and minimum values off � x � y � � xy on the ellipse 5x2 � y2 � 7.maximum value =minimum value =

14.(1 pt) You are hiking the Inca Trail on the way to MachuPiecho. When you arrive at the hightest point on the trail, whichof the following are possibilities? In alphabetical order withoutpunctuation or spacing, list the letters which indicate possibili-ties.

(A) The path passes through the center of a set of concentriccontour lines.(B) The path is tangent to a contour line.(C) The path follows a contour line.(D) The path crosses a contour line.possibilities:


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VMultIntegrals1Double due 4/3/08 at 2:00 AM.

1.(1 pt) Consider the solid that lies above the square R �� 0 � 2 � � � 0 � 2 � and below the elliptic paraboloid z � 100 � x2 � 1y2.

(A) Estimate the volume by dividing R into 4 equal squaresand choosing the sample points to lie in the lower left hand cor-ners.

(B) Estimate the volume by dividing R into 4 equal squaresand choosing the sample points to lie in the upper right handcorners..

(C) What is the average of the two answers from (A) and (B)?

(D) Using iterated integrals, compute the exact value of thevolume.

2.(1 pt) Evaluate the iterated integral� 1

0

� 3

012x2y3 dxdy

3.(1 pt) Evaluate the iterated integral� 5

4

� 2

1� 4x � y � � 2 dydx

4.(1 pt) Calculate the double integral� �

R� 10x � 8y � 80 � dA

where R is the region: 0�

x�

4 � 0�

y�

5.

5.(1 pt) Calculate the double integral� �

Rxcos � 1x � y � dA

where R is the region: 0�

x� 2π

6 � 0�

y� 2π

4

6.(1 pt) Calculate the volume under the elliptic paraboloidz � 4x2 � 7y2 and over the rectangle R � � � 2 � 2 � � � � 1 � 1 � .

7.(1 pt) Using geometry, calculate the volume of the solidunder z � �

1 � x2 � y2 and over the circular disk x2 � y2 �1.

8.(1 pt) Using the maxima and minima of the function, pro-duce upper and lower estimates of the integral

I � � �D

e7�x2 � y2 � dA where D is the circular disk: x2 � y2 �

3.�I

�

9.(1 pt) Evaluate the iterated integral I � � 1

0

� 1�

x

1 � x� 24x2 �

8y � dydx

10.(1 pt) Evaluate the double integral I � � �D

xydA where D

is the triangular region with vertices � 0 � 0 � � � 6 � 0 � � � 0 � 2 � .11.(1 pt) Find the volume of the solid bounded by the planes

x = 0, y = 0, z = 0, and x + y + z = 1.

12.(1 pt) Evaluate the integral by reversing the order of inte-gration.� 1

0

� 5

5yex2

dxdy �13.(1 pt) Match the following integrals with the verbal de-

scriptions of the solids whose volumes they give. Put the letterof the verbal description to the left of the corresponding integral.

1.� 1

� 1

� �1 � x2

� �1 � x2

1 � x2 � y2 dydx

2.� 2

0

� 2

� 2

�4 � y2 dydx

3.� 1

0

�� y

y24x2 � 3y2 dxdy

4.� 1�

3

0

� 12

�1 � 3y2

0

�1 � 4x2 � 3y2 dxdy

5.� 2

� 2

� 4� �

4 � x2

44x � 3y dydx

A. Solid bounded by a circular paraboloid and a plane.B. One half of a cylindrical rod.C. Solid under a plane and over one half of a circular disk.D. Solid under an elliptic paraboloid and over a planar re-

gion bounded by two parabolas.E. One eighth of an ellipsoid.

14.(1 pt) If� 0

� 1f � x � dx � � 2 and

� 5

2g � x � dx � � 1, what is

the value of� �

Df � x � g � y � dA where D is the square: � 1

�x

�0 � 2

�y

�5?


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VMultIntegrals2Polar due 4/4/08 at 2:00 AM.

1.(1 pt)

Using polar coordinates, evaluate the integral� �

Rsin � x2 �

y2 � dA where R is the region 1�

x2 � y2 �25.

2.(1 pt)Using polar coordinates, evaluate the integral which gives

the area which lies in the first quadrant between the circlesx2 � y2 � 400 and x2 � 20x � y2 � 0.

3.(1 pt)Use the polar coordinates to find the volume of a sphere of

radius 9.

4.(1 pt)A cylindrical drill with radius 5 is used to bore a hole

throught the center of a sphere of radius 7. Find the volumeof the ring shaped solid that remains.

5.(1 pt)A. Using polar coordinates, evaluate the improper integral� �R2

e � 8�x2 � y2 � dx dy.

B. Use part A to evaluate the improper integral� ∞

� ∞e � 8x2

dx.

6.(1 pt)A sprinkler distributes water in a circular pattern, supplying

water to a depth of e � r feet per hour at a distance of r feet fromthe sprinkler.

A. What is the total amount of water supplied per hour insideof a circle of radius 14?

f t3 � hB. What is the total amount of water that goes throught the

sprinkler per hour?f t3 � h


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VMultIntegrals3Appl due 4/5/08 at 2:00 AM.

1.(1 pt)Electric charge is distributed over the disk

x2 � y2 �1 so that the charge density at (x,y) is σ � x � y � �

2 � x2 � y2 coulombs per square meter.Find the total charge on the disk.

2.(1 pt)A lamina occupies the part of the disk x2 � y2 �

4 in the firstquadrant and the density at each point is given by the functionρ � x � y � � 4 � x2 � y2 � .

A. What is the total mass?B. What is the moment about the x-axis?C. What is the moment about the y-axis?D. Where is the center of mass? ( , )E. What is the moment of inertia about the origin?

3.(1 pt)A lamp has two bulbs, each of a type with an average lifetime

of 10 hours. The probability density function for the lifetime ofa bulb is f � t � � 1

10 e � t 10 � t �0.

What is the probability that both of the bulbs will fail within 4hours?

4.(1 pt)You are getting married and your dearest relative has baked

you a cake which fills the volume between the two planes, z � 0and z � 9x � 2y � c, and inside the cylinder x2 � y2 � 1. Youare to cut it in half by making two vertical slices from the centeroutward. Suppose one of the slices is at θ � 0 and the other is atθ � ψ.

What is the limit, limc � ∞

ψ?


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VMultIntegrals4Surface due 4/6/08 at 1:00 AM.

1.(1 pt) Find the surface area of the part of the plane 1x �3y � z � 2 that lies inside the cylinder x2 � y2 � 1.

2.(1 pt) Find the surface area of the part of the circular parab-oloid z � x2 � y2 that lies inside the cylinder x2 � y2 � 25.

3.(1 pt) The vector equation r � u � v � � ucosvi � usinvj � vk,0

�v

�7π, 0

�u

�1, describes a helicoid (spiral ramp). What

is the surface area?

4.(1 pt) Find the surface area of the surface of revolution gen-erated by revolving the graph y � x3, 0

�x

�9 around the x-

axis.


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VMultIntegrals5Triple due 4/7/08 at 2:00 AM.

1.(1 pt) Evaluate the triple integral� � �E

xyzdV

where E is the solid: 0�

z�

7, 0�

y�

z, 0�

x�

y.

2.(1 pt) Find the volume of the solid enclosed by theparaboloids z � 16 � x2 � y2 � and z � 8 � 16 � x2 � y2 � .

3.(1 pt) Find the average value of the function f � x � y � z � �x2 � y2 � z2 over the rectangular prism 0

�x

�2, 0

�y

�4,

0�

z�

2

4.(1 pt) Find the mass of the rectangular prism 0�

x�

3,0

�y

�3, 0

�z

�1, with density function ρ � x � y � z � � x. You

might find formula No. 13 on page 1014 of the text helpful.

5.(1 pt) Use cylindrical coordinates to evaluate the triple in-

tegral� � �

E

�x2 � y2 dV , where E is the solid bounded by the

circular paraboloid z � 16 � 9 � x2 � y2 � and the xy -plane.

6.(1 pt) Use spherical coordinates to evaluate the triple inte-

gral� � �

Ex2 � y2 � z2 dV , where E is the ball: x2 � y2 � z2 �

1.

7.(1 pt)Match the integrals with the type of coordinates which make

them the easiest to do. Put the letter of the coordinate system tothe left of the number of the integral.

1.� 1

0

� y2

0

1x

dx dy

2.� � �

Ez dV where E is: 1

�x

�2 � 3

�y

�4 � 5

�z

�6

3.� � �

EdV where E is: x2 � y2 � z2 �

4 � x � 0 � y �0 � z � 0

4.� �

D

1x2 � y2 dA where D is: x2 � y2 �

4

5.� � �

Ez2 dV where E is: � 2

�z

�2 � 1

�x2 � y2 �

2

A. cartesian coordinatesB. polar coordinatesC. cylindrical coordinatesD. spherical coordinates

8.(1 pt) A volcano fills the volume between the graphs z � 0and z � 1� x2 � y2 � 12 , and outside the cylinder x2 � y2 � 1. Find the

volume of this volcano.


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Tom Robbins WW Prob Lib2 Summer 2001Sample WeBWorK problems. WeBWorK assignment VectorCalculus1 due 5/3/08 at 2:00 AM.

1.(1 pt)Consider the transformation T : x � 14

50 u � 4850 v � y � 48

50 u � 1450 v

A. Compute the Jacobian:∂�x � y �

∂�u � v � �B. The transformation is linear, which implies that it trans-

forms lines into lines. Thus, it transforms the square S : � 50�

u�

50 �� 50�

v�

50 into a square T � S � with vertices:T(50, 50) = ( , )T(-50, 50) = ( , )T(-50, -50) = ( , )T(50, -50) = ( , )C. Use the transformation T to evaluate the integral� �

T�S � x2 � y2 dA

2.(1 pt)Compute the gradient vector fields of the following func-

tions:A. f � x � y � � 7x2 � 1y2

∇ f � x � y � � i � jB. f � x � y � � x10y5 �

∇ f � x � y � � i � jC. f � x � y � � 7x � 1y

∇ f � x � y � � i � jD. f � x � y � z � � 7x � 1y � 10z

∇ f � x � y � � i � j � kE. f � x � y � z � � 7x2 � 1y2 � 10z2

∇ f � x � y � z � � i � j � k

3.(1 pt)Match the following vector fields with the verbal descriptions

of the level curves or level surfaces to which they are perpendic-ular by putting the letter of the verbal description to the left ofthe number of the vector field.

1. F � yi � xj2. F � xi � yj � zk3. F � xi � yj4. F � � yi � xj5. F � 2xi � yj � zk6. F � xi � yj � zk7. F � 2i � j � k8. F � xi � yj � k9. F � 2i � j

10. F � xi � yj11. F � 2xi � yj

A. planesB. hyperbolasC. ellipsoidsD. ellipsesE. paraboloidsF. lines

G. hyperboloidsH. circlesI. spheres

4.(1 pt)Compute the total mass of a wire bent in a quarter circle with

parametric equations: x � 7cost � y � 7sin t � 0�

t� π

2 and den-sity function ρ � x � y � � x2 � y2.

5.(1 pt) Let C be the curve which is the union of two line seg-ments, the first going from (0, 0) to (3, -4) and the second goingfrom (3, -4) to (6, 0).

Computer the line integral�

C3dy � 4dx.

6.(1 pt)Let F be the radial force field F � xi � yj. Find the work

done by this force along the following two curves, both whichgo from (0, 0) to (4, 16). (Compare your answers!)

A. If C1 is the parabola: x � t � y � t2 � 0�

t�

4, then�C1

F � dr �B. If C2 is the straight line segment: x � 4t2 � y � 16t2 � 0

�t

�1, then

�C2

F � dr �7.(1 pt)Let C be the counter-clockwise planar circle with center at

the origin and radius r � 0. Without computing them, deter-mine for the following vector fields F whether the line integrals�

CF � dr are positive, negative, or zero and type P, N, or Z as

appropriate.A. F = the radial vector field = xi � yj:B. F = the circulating vector field = � yi � xj:C. F = the circulating vector field = yi � xj:D. F = the constant vector field = i � j:

8.(1 pt)Consider a wire in the shape of a helix r � t � � 7costi �

7sin tj � 1tk � 0 �t

�2π with constant density function

ρ � x � y � z � � 1.A. Determine the mass of the wire:B. Determine the coordinates of the center of mass: ( ,

, )C. Determine the moment of inertia about the z-axis:

9.(1 pt)Find the work done by the force field F � x � y � z � � 6xi � 6yj �

7k on a particle that moves along the helix r � t � � 1cos � t � i �1sin � t � j � 4tk � 0 �

t�

2π.

10.(1 pt)1

A curve C is given by a vector function r � t � � 5 �t

�11, with

unit tangent T � t � , unit normal N � t � , and unit binormal B � t � . In-dicate whether the following line integrals are positive, negative,or zero by typing P, N, or Z as appropriate:

A.�

CT � dr �

B.�

CN � dr �

C.�

CB � dr �

11.(1 pt)

Suppose that� �

Df � x � y � dA � 1 where D is the disk x2 �

y2 �16. Now suppose E is the disk x2 � y2 �

16 and g � x � y � �2 f � x

1 � y1� . What is the value of

� �E

g � x � y � dA?

12.(1 pt)A lattice point in the plane is a point (a, b) with both coor-

dinates equal to integers. For example, (-1, 2) is a lattice pointbut (1/2, 3) is not. If D(R) is the disk of radius R and center theorigin, count the lattice points inside D(R) and call this number

L(R). What is the limit, limR � ∞

L � R �R2 ?


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1.(1 pt) For each of the following vector fields F , decidewhether it is conservative or not by computing curl F . Type ina potential function f (that is, ∇ f � F). If it is not conservative,type N.

A. F � x � y � �� 10x � 5y � i � � � 5x � 4y � jf � x � y � �

B. F � x � y � � 5yi � 6xjf � x � y � �

C. F � x � y � z � � 5xi � 6yj � kf � x � y � z � �

D. F � x � y � �� 5siny � i � � � 10y � 5xcosy � jf � x � y � �

E. F � x � y � z � � 5x2i � 5y2j � 2z2kf � x � y � z � �

Note: Your answers should be either expressions of x, y andz (e.g. “3xy + 2yz”), or the letter “N”

2.(1 pt) If C is the curve given by r � t � � � 1 � 1sint � i �� 1 � 3sin2 t � j � � 1 � 4sin3 t � k, 0

�t

� π2 and F is the radial

vector field F � x � y � z � � xi � yj � zk, compute the work done byF on a particle moving along C.

3.(1 pt) Suppose C is any curve from � 0 � 0 � 0 � to � 1 � 1 � 1 � andF � x � y � z � � � 2z � 3y � i � � 4z � 3x � j � � 4y � 2x � k. Compute theline integral

�C F � dr.

4.(1 pt) Let F � x � y � � � yi�

xjx2 � y2 and let C be the circle r � t � �� cost � i � � sin t � j, 0

�t

�2π.

A. Compute ∂Q∂x

Note: Your answer should be an expression of x and y; e.g. ”3xy- y”

B. Compute ∂P∂y

Note: Your answer should be an expression of x and y; e.g. ”3xy- y”

C. Compute�C F � dr

Note: Your answer should be a numberD. Is F conservative? Type Y if yes, type N if no.

5.(1 pt) Determine whether the given set is open, connected,and simply connected. For example, if it is open, connected, butnot simply connected, type ”YYN” standing for ”Yes, Yes, No.”

A.� � x � y � �

x � 1 � y � 2 �

B.� � x � y � �

2x2 � y2 � 1 �C.

� � x � y � �x2 � y2 � 1 �

D.� � x � y � �

x2 � y2 � 1 �E.

� � x � y � �1 �

x2 � y2 � 4 �

6.(1 pt) Let C be the positively oriented circle x2 � y2 � 1.Use Green’s Theorem to evaluate the line integral

�C 11ydx �

13xdy.

7.(1 pt) Let C be the positively oriented square with vertices� 0 � 0 � , � 2 � 0 � , � 2 � 2 � , � 0 � 2 � . Use Green’s Theorem to evaluate theline integral

�C 7y2xdx � 9x2ydy.

8.(1 pt) Find a parametrization of the curve x2 3 � y2 3 � 1and use it to compute the area of the interior.

9.(1 pt) Let F � 4xi � 5yj � 1zk. Compute the divergence andthe curl.

A. div F �B. curl F � i � j � k

10.(1 pt) Let F � � 6yz � i � � 10xz � j � � 7xy � k. Compute thefollowing:

A. div F �B. curl F � i � j � kC. div curl F �Note: Your answers should be expressions of x, y and/or z;

e.g. ”3xy” or ”z” or ”5”

11.(1 pt) Let F be any vector field of the form F � f � x � i �g � y � j � h � z � k and let G be any vector field of the form F �f � y � z � i � g � x � z � j � h � x � y � k. Indicate whether the followingstatements are true or false by placing ”T” or ”F” to the left ofthe statement.

1. G is incompressible2. G is irrotational3. F is incompressible4. F is irrotational

12.(1 pt) Let F � � 3yi � 5xj. Use the tangential vector formof Green’s Theorem to compute the circulation integral

�C F � dr

where C is the positively oriented circle x2 � y2 � 25.

13.(1 pt) Let F � 3xi � 4yj and let n be the outward unit nor-mal vector to the positively oriented circle x2 � y2 � 16. Com-pute the flux integral

�C F � nds.

14.(1 pt) A rock with a mass of 3 kilograms is put aboardan airplane in New York City and flown to Boston. How muchwork does the gravitational field of the earth do on the rock?

Newton-meters15.(1 pt) Suppose F � F � x � y � z � is a gradient field with F �

∇ f , S is a level surface of f, and C is a curve on S. What is thevalue of the line integral

�C F � dr?

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16.(1 pt)A vector field gives a geographical description of the flow of

money in a society. In the neighborhood of a political conven-tion, the divergence of this vector field is:

� A. negative� B. zero� C. positive


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1.(1 pt) Evaluate� �

S

�1 � x2 � y2 dS where S is the heli-

coid: r � u � v � � ucos � v � i � usin � v � j � vk, with 0�

u�

3 � 0 �v

�4π

2.(1 pt) Find the surface area of the part of the spherex2 � y2 � z2 � 9 that lies above the cone z � �

x2 � y2

3.(1 pt) A fluid has density 5 and velocity field v � � yi �xj � 3zk.Find the rate of flow outward through the sphere x2 � y2 � z2 � 9

4.(1 pt) Let S be the part of the plane 2x � 1y � z � 3 whichlies in the first octant, oriented upward. Find the flux of thevector field F � 3i � 3j � 2k across the surface S.

5.(1 pt) Use Gauss’s law to find the charge enclosed bythe cube with vertices �� 1 � � 1 � � 1 � if the electric field isE � x � y � z � � 1xi � 4yj � 2zk.

ε0

6.(1 pt) The temperature u in a star of conductivity 7 isinversely proportional to the distance from the center: u �

7�x2 � y2 � z2

.

If the star is a sphere of radius 1, find the rate of heat flow out-ward across the surface of the star.

7.(1 pt) Use Stoke’s theorem to evaluate� �

ScurlF � dS where

F � x � y � z � � � 14yzi � 14xzj � 8� x2 � y2 � zk and S is the part of theparaboloid z � x2 � y2 that lies inside the cylinder x2 � y2 � 1,oriented upward.

8.(1 pt) Use Stoke’s Theorem to evaluate�

CF � dr where

F � x � y � z � � xi � yj � 3� x2 � y2 � k and C is the boundary of the

part of the paraboloid where z � 81 � x2 � y2 which lies abovethe xy-plane and C is oriented counterclockwise when viewedfrom above.

9.(1 pt) Use the divergence theorem to find the outwardflux of the vector field F � x � y � z � � 1x2i � 2y2j � 2z2k across theboundary of the rectangular prism: 0

�x

�3 � 0 �

y�

4 � 0 �z

�1.

10.(1 pt) If a parametric surface given by r1 � u � v � � f � u � v � i �g � u � v � j � h � u � v � k and � 2

�u

�2 �� 2

�v

�2, has surface area

equal to 4, what is the surface area of the parametric surfacegiven by r2 � u � v � � 2r1 � u � v � with � 2

�u

�2 �� 2

�v

�2?

11.(1 pt) Suppose F is a radial force field, S1 is a sphere of ra-

dius 7 centered at the origin, and the flux integral� �

S1

F � dS �3.Let S2 be a sphere of radius 49 centered at the origin, and con-

sider the flux integral� �

S2

F � dS.

(A) If the magnitude of F is inversely proportional to thesquare of the distance from the origin,what is the value of� �

S2

F � dS?

(B) If the magnitude of F is inversely proportional to the cube

of the distance from the origin, what is the value of� �

S2

F � dS?

12.(1 pt)In springtime, the average value over time of the divergence

of the vector field which represents air flow is:� A. zero� B. negative� C. positive


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tom robbins ww prob lib2 summer 2001 - math

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