I. Let f (x, y) be a differentiable function such that ∇f = hey z 2 , xey z 2 , 2xey zi 1. Find Du f (1, 0, −1) if ~u is the unit vector in the direction of h−2, 1, 2i. 2. Determine the divergence of ∇f at (1, 0, −1). II. Show that the only critical point of g(x, y) = x1 + xy − y8 is (−1/2, 4). Determine whether (−1/2, 4) is a relative maximum, relative minimum or neither. III. Evaluate the following integrals. Z 1. 0

1

1

Z

√ 3x

1 p

y4

+3

Z

dydx

2

Z √4−y2 Z √4−x2 −y2

2. 0

0

2 +y 2 +z 2 )3/2

e(x

dzdxdy

0

IV. Find the amount of work done by the force F~ (x, y) = xy 2~i + (2y − x2 )~j in moving an object along the arc of y = x2 from (−1, 1) to (1, 1). H V. Let C be the circle x2 + y 2 = 1. Evaluate C (sinh x − yx2 )dx + (xy 2 + sin−1 y)dy. VI. Given F~ (x, y) = h2xy − cos y, x2 + x sin y + 2i. 1. Find all functions f such that ∇f = F~ R 2. Evaluate C F~ · d~r where C is any smooth curve from (1, 0) to (2, π). VII. Let Ω be the portion of the plane 3x + 2y + 6z = 12 which lies in the first octant. Set up an RR iterated double integral in Cartesian coordinates equal to Ω (7x + 5y + 12z − 24)dS. VIII. Determine whether the series converges or diverges. ∞ X n 1. ln n n=2

2.

∞ X

(−1)n

n=1

3n 1 n

+2

2n

IX. Find the radius and interval of convergence of the power series

∞ X xn 3n n=1

X. Given that

1 1−x

=

P∞

n n=0 x , show that

x 1+x

=

P∞

n3

.

n n+1 . Use this to find the sum n=0 (−1) x

P∞

n n+1 n=0 (−1) 3n+1 .

Math 55 sample final exam 2