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Test II Questions for Calculus III - Spring 2008 | MTH 253, Exams of Advanced Calculus

Portland Community CollegeAdvanced Calculus

Material Type: Exam; Class: Calculus III; Subject: Math; University: Portland Community College; Term: Spring 2008;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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MTH 253 - Spring Term 2008
Test 2 - Calculator Portion
Given May 20, 2008 Name
All work on this test will be evaluated for your style of presentation as well as for the "correctness" of your "answer."
Follow the writing guidelines established during lecture and spelled out in your Test 2 guide. To receive full credit all
relevant algebra steps must be shown on the paper.
1. Write down (through the 4th digit after the decimal point) each partial sum ( 123
,,,SSS…) of
the series
()
()
1
2
1
1
!
k
kk
+
∞
=
−
∑ until you have established the value of
()
()
1
2
1
1
!
k
kk
+
∞
=
−
∑ accurate through
the 3rd digit after the decimal point. State this value and how you know that you’ve established
said value. (12 points)
pf3
pf4
pf5
pf8

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Download Test II Questions for Calculus III - Spring 2008 | MTH 253 and more Exams Advanced Calculus in PDF only on Docsity!

MTH 253 - Spring Term 2008 Test 2 - Calculator Portion Given May 20, 2008 Name

All work on this test will be evaluated for your style of presentation as well as for the "correctness" of your "answer." Follow the writing guidelines established during lecture and spelled out in your Test 2 guide. To receive full credit all relevant algebra steps must be shown on the paper.

  1. Write down (through the 4 th^ digit after the decimal point) each partial sum ( S 1 (^) , S 2 (^) , S 3 ,…) of

the series

1

1 2

k

k k

∞^ +

∑ until you have established the value of^

1

1 2

k

k k

∞^ +

∑ accurate through

the 3 rd^ digit after the decimal point. State this value and how you know that you’ve established said value. (12 points)

2. The 0 th, 1st^ , and 2 rd^ derivatives of f ( x ) 31

x

= are shown in Table 1. Find, without using the Taylor command on your calculator , the second degree Taylor polynomial for f centered at 8. Your formula should have no decimals in it. (10 points)

Table 1

k f (^ k )^ ( x )

(^0) x −1/

− x −

x −

  1. There are functions called the hyperbolic sine function ( sinh ) and hyperbolic cosine function ( cosh ). While there are similarities between these functions and the sine and cosine functions, there are differences as well.

One similarity is that cosh 0( ) = cos 0( )= 1 and sinh 0( ) = sin 0( )= 0.

As far as derivatives go, there are similarities (^ d^ ( sinh ( x )) cosh ( x ))

dx = and slight

differences (^ d^ ( cosh ( x )) sinh( x )

dx

Use Taylor’s formula to develop the Maclaurin series for sinh ( x ). Make sure that your work is

clearly outlined and that your formula is fully developed from Taylor’s formula. (10 points)

  1. Determine and state whether each given series is absolutely convergent, conditionally convergent, or divergent. Simply state your conclusion – no other work should be shown. (You may want to do some work on your scratch paper to help you decide your answer.) (14 points total)

` ( )

1

1 k k k

∞

∑ is^.

` ( )

1 1.

1 k k k

∞

∑ is^.

( )^3

1 2

k k k

∞

∑ is^.

k 1^ k

k^ π

∞

∑ is^.

1

sin 2 k

k

k

∞

∑ is^.

1

ln^1 k k

∞

∑ ⎜⎝ ⎟⎠ is^.

1

k k k k

∞

is.

5. Find the interval of convergence for the series (^2 )

1

k k k

x k

∞

. (14 points)

6. Find the interval of convergence for the series (^ )

1

k k k

∞ x k

∑.^ (14 points)