Department of Mathematics

University of Pittsburgh

MATH 0230 (Calculus II)

Midterm 2, Fall 2017

Instructor: Kiumars Kaveh

Last Name: Student Number:

First Name:

TIME ALLOWED: 50 MINUTES. TOTAL MARKS: 50NO AIDS ALLOWED.WRITE SOLUTIONS ON THE SPACE PROVIDED.PLEASE READ THROUGH THE ENTIRE TEST BEFORE STARTINGAND TAKE NOTEOF HOWMANY POINTS EACHQUESTION ISWORTH.FOR FULLMARKYOUMUST PRESENT YOUR SOLUTION CLEARLY.

Question Mark

1 /10

2 /10

3 /8

4 /12

5 /10

6 /1

TOTAL /50 + 1

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Kaveh 1

Aydin

1.[10 points] Consider the curve in the plane given parametrically by:

x(t) = sin(t), y(t) = tan(t).

(a) [6 points] Find the (parametric) equation of tangent line to the curveat the point t = ⇡/3.

(b) [4 points] Setup an integral that represents the length of this curvefrom t = 0 to t = ⇡.

2

(a)xtt ) = Cost ) y 'ct ) =

seict )

xkng )=tz y 'cB)= 4

× ( B) = 53,2 yens ) =53

.

E9u . of tangent line at t.mg :

Xcti = tzt + I3{ yet ,=4t+r .

M(b)length = fcaiit.se#dt .

o

2.[10 points] Find the area of the region that lies inside the polar curver = 1� sin(✓) and outside the polar curve r = 1.

3

Area =

0.1€ asthenia - ( hafner'E,

the:c)rct ) =

I - Sin ( A )

§€- Sino ))2da = §In - sinca ) + singed da =

217

217

aerial,

+ ¥ ( × - Sindy= tz ( 217-17 ) t n

= Is + 2 + If =Zt 3¥ .

Area = (2+342) - Iz = 2+4 .

3.[12 points]

(a) [3 points] Show that the seriesP1

n=12+sin(2n)

n2 is absolutely convergent.

(b) [3 points] Determine if the seriesP1

n=1(�1)np

n is convergent. In the caseof convergence, determine if it is also absolutely convergent.

(c) [3 points] Find the sum of the seriesP1

n=22n

3n .

(d) [3 points] Find the sum of the seriesP1

n=3 1/n(n� 2).

4

ybecause

.us?Yyg#

noteofFILMY

'

§

12+51*1s

§T3zbut Enzt is com .

so by comparison test §2tsnind2nL is abs .Conn

.

o< ¥ & decreasing seq . so by ( Leibnitz ) alt . series test

§ t is Yin is corn . But §tn is not com ( for example

by integral test ) . so it is not abs .

Com .

⇐÷÷E#n=÷÷=÷⇒.

⇐ ÷ - c÷5n⇐#n= ÷ '3=÷ '

2

2- = 1=2 - f- ( partial fraction )

h ( n - 2)

Is # = Inez. it = 't .

'she . ⇒ tests

+

#t*s+...

'" = 1+21=3 .

4.[8 points]

(a) [4 points] Determine the radius of convergence of the power seriesP1n=0

xn

n2n .

(b) [4 points] Find the power series representation of the function 1+x2

1+3x .

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(a)Root test : nhjma"F×÷nl=tI3n¥z cabs ,

= ¥ . By root test 1×1<1 seriescom .

& 1¥ > , seriesdive so Yxls}d9Y

⇒ radius of conv . = 2 .

a

(b) # = [ x"

n =:¥ = [ C- 3×5

n=O

1-1×2 A

#= [ (4×2)+3×1

"

h=0

A na

= [ C-3)"

×n+2+[ t3× )oo

n=o" ⇒

¢.si#Yx?=nEcs352xn+Ioaxsn=itt3sx+nE

5.[10 points] Consider the function f(x) = 1/(1� x)2.

(a) [4 points] Find the Taylor polynomial T2(x) at x = 0 of f(x).

(b) [4 points] Find the Taylor series T (x) of f(x).

(c) [2 points] Find the radius of convergence of the power series T (x).

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fix)= ( y. x )'

2)'

=2 ( i -xp-4

f-"

( × )=(

2×3 ) ( 1 - × )

:;

- ( nt2 )

f( "

=( nty ! ( 1 - × )

f-( "

( o ) = ( nti ) !

(a) Tzcx ) =fco ) +

f{o)× + f€)×22 !

= I + ZX + 3×2

( b ) 1- ( × ) = I +2×+3×4 4×3-1 , . .

a n

= [ (ntD×

n=O

(C) By root test radius of Convergence is z .

6.[1 point] Draw a cartoon of yourself eating a whole turkey!

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