## [answered] Math 2224 Common Final Exam Fall 2012 FORM C INSTRUCTIONS:

Can you help me with number 3 please.

I've been stuck on this for so long

Math 2224 Common Final Exam Fall 2012 FORM C

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Student ID #  Consider the series S = 

? 

X

?3 n

n=1 2 . Which of the following statements is true?  50 1) The 50-th partial sum equals

3) S = ? 3 3

5 2 ? 3

5 2) S = 3

5 4) The 50-th partial sum diverges  The interval of convergence of

1) (?3, 5) 2

5 2) x = 1 only ?

X n! (x ? 1)n

equals

4n

n=1

3) (?1, 1) 1 4) (0, 2)  The value of 0 1) e2 ? 1 ? x 2)  Evaluate ey

dy dx is given by

y2 Z 4Z 2 1 2

e ?1

2 3) e2 4) 1 2

e ?1

4 x2 + y 2 ? x 4

or state that the limit does not exist

(x,y)?(0,0) 2(x2 + y 2 )

lim 1) 0 2) The limit does not exist  The sequence an = 3) 1

2 4) 1

4 ?

?

n2 + 4n ? n2 + 1 1) converges to 2 2) converges to ?1 3) diverges 4) converges to 0  If z = x3 ? y, where x = tet and y = sin(t), then dz

equals

dt 1) 3t2 e3t (t + 1) ? cos(t) 2) 3x2 et (t + 1) + cos(t) 3) 3t2 e3t + cos(t) 4) 3x2 et ? cos(t)  The volume of the pyramid with vertices (0, 0, 0), (1, 0, 0), (1, 2, 0), (0, 2, 0), and (1, 2, 3) is

represented by which of the following integrals? 1) Z 3 Z 2z/3 Z z/3

0 3) 0 0 2) 0 Z 2 Z 1 Z 3y/2

0 dx dy dz 0 Z 2Z 1Z 3

0 dz dx dy 4) 0 Z 3Z 2

0 dz dx dy 0 2z/3 Z 1

z/3 2 dx dy dz  Which of the following surfaces matches the level curves below?

1) z = x2 ? y 2 + 6

3) z = q 2) z 2 = x + y + 6 x2 + y 2 + 6 4) z = q x2 + y 2 4

3

2 y 1 0

?1

?2 ?3

?4

?4 ?3 ?2 ?1 0 1 2 3 4 x  The Maclaurin series of f (x) = e?2x equals

1) ?

X (?1)n (?2x)2n

(2n)!

n=0

?

X xn

3)

n=0 n! 2) 4) ?

X (?1)n (?2x)2n+1

(2n + 1)!

n=0

?

X (?2x)n

n!

n=0  At the point (0, 2), the function f (x, y) = x3 + x2 y + y 2 ? 4y + 2 has

1) not a critical point 2) a saddle point 3) a local minimum 4) a local maximum 3 2

2

 The volume of the region

q that is above the xy?plane, inside the surface x + y = 4 and

beneath the surface z = x2 + y 2 is represented by which of the following integrals in spherical

coordinates? 1) Z 2? Z ?/4 Z 2/ sin ?

0 3) 0 ?/4 ? sin ? d? d? d? 2) Z 2? Z ?/2 Z 2/ sin ? 0 0 Z 2? Z ?/2 Z 2

0 2 ?2 sin ? d? d? d? 4) 0 0 ?2 sin ? d? d? d? 0 Z 2? Z ?/4 Z 2 0  The series ?/4 ?2 sin ? d? d? d? 0 ?

X (?1)2n

converges by the

2

n=1 n + 1 1) direct comparison test, using bn = 1/n2 2) ratio test 3) n-th term test since n??

lim an = 0 4) alternating series test 1

dA, where D is the region in the first quadrant bounded

D 1 + x2 + y 2

by y = x, x = 0, and x2 + y 2 = 4, is given by  The double integral 1) ?

tan?1 (2)

4 Z Z 2) ?

tan?1 (4)

4 3) ?

ln 5

4  Find the directional derivative of the function f (x, y, z) =

direction of the vector v = 4i ? 2j + 4k

1) 3

2 2) 2 3) 1

3 4) 4 1

4 4) ? ?

ln 5

8 xyz at the point P (2, 1, 2) in the

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