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Final Exam with Answer Key and Remarks - Calculus II | MAC 2312, Exams of Calculus

Material Type: Exam; Professor: Hudson; Class: Calculus II; Subject: Mathematics Calculus and Precalculus; University: Florida International University; Term: Summer 2004;

Typology: Exams

Pre 2010

Uploaded on 03/10/2009

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MAC 2312 August 12, 2004
Final Exam Prof. S. Hudson
1) (10pts) For each series, answer either Converges absolutely (CA) or Con-
verges conditionally (CC) or Diverges (D). State which “test” you are using
and show work.
a) P
k=1(1)k(1 + k)2ln(k)
b) P
k=1
(1)k
2k+3
2) (5pts) Set up and simplify the integral with which you would find the
length of the curve y=1
3(x2+2)3/2from x= 0 to x= 3. Do not evaluate.
3) (5pts) Express 1 2/3+4/58/7 + 16/9 in sigma notation.
4) [5 pts each] Compute (or explain why no answer exists):
Rtan2xsec2x dx
Rx2exdx
Rπ
0|cos(x)|dx =
R+
e
dx
xln x
5) (15pts) a) Sketch the rose r=sin2θ.
b) Write down, but do not evaluate, an integral for the area of region
enclosed by the rose.
c) Use parametric equations to find dy/dx for this curve.
6) (10pts) a) Find the Taylor Series for the function f(x) = sin(x)
xat x= 0.
b) Use the result in (a) to approximate the integral R1
0sin(x)/x dx (include
3 nonzero terms).
1
pf3

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MAC 2312 August 12, 2004 Final Exam Prof. S. Hudson

  1. (10pts) For each series, answer either Converges absolutely (CA) or Con- verges conditionally (CC) or Diverges (D). State which “test” you are using and show work.

a)

k=1(−1)

k(1 + k)− (^2) ln(k)

b)

k=

(−1)k 2 k+

  1. (5pts) Set up and simplify the integral with which you would find the length of the curve y = 13 (x^2 +2)^3 /^2 from x = 0 to x = 3. Do not evaluate.

  2. (5pts) Express 1 − 2 /3 + 4/ 5 − 8 /7 + 16/9 in sigma notation.

  3. [5 pts each] Compute (or explain why no answer exists):

∫ tan^2 x sec^2 x dx ∫ x^2 ex^ dx ∫ (^) π 0 |^ cos(x)|^ dx^ = ∫ (^) +∞ e

dx x ln x

  1. (15pts) a) Sketch the rose r = sin 2 θ.

b) Write down, but do not evaluate, an integral for the area of region enclosed by the rose.

c) Use parametric equations to find dy/dx for this curve.

  1. (10pts) a) Find the Taylor Series for the function f (x) = sin( xx ) at x = 0.

b) Use the result in (a) to approximate the integral

0 sin(x)/x dx^ (include 3 nonzero terms).

  1. (5pts) Find the interval of convergence of the series.

∑^ ∞

k=

2 kxk k

  1. [10 pts] Choose ONE:

a) State and prove the integration formula for area in polar coordi- nates. Draw a picture showing α, β, etc., and use a

and a limit in your explanation.

b) Prove: if a power series converges to f on (x 0 − R, x 0 + R), then it is the Taylor series of f at x 0. (Prove that ck =.. ., as Taylor predicts).

  1. [20 pts] Answer True or False:

The shell method formula for volume is V =

∫ (^) b a 2 πxf^ (x)^ dx.

The formula for arc length in Ch. 6 is L =

∫ (^) b a

1 + (f ′(x))^2 dx.

The integral

1 1 /x

p (^) dx converges if and only if p > 1.

There are exactly 8 points on the graph of r = cos(2θ) where the tangent line is horizontal.

If 0 ≤ an ≤ bn and

bn converges, then

an converges.

The Mclaurin series for tan−^1 x is x − x^3 /3 + x^5 /5 +... but it doesn’t converge for all real x.

The polar equations r = sin 4θ and r = cos 4θ have the same graphs.

A bounded increasing sequence must converge.

The sequence 10

n n! (for^ n^ = 0,^1 ,^2.. .) is strictly decreasing.

The series 1 − 1 /2 + 1/ 3 − 1 /4 +... converges conditionally to ln 2.