MATHEMATICS 55 (Elementary Analysis III) Problem Set

Due: October 7, 1:00 PM MB 238

DIRECTIONS: Show all necessary solutions and box final answers. Use only black or blue ink pens. Good luck! I. Give f (x, y) = x2 + 2y 2 − x2 y 1. Find the directional derivative of f at (2, −1) along h−2, 1i.

(3 pts)

2. Determine and classify all the critical points of f .

(4 pts)

II. Find the dimensions of the rectangular box of maximum volume that can be inscribed in the ellipsoid x2 y2 z2 + + = 1 using Lagrange multipliers. (5 pts) a2 b2 c2 III. Use triple integral to find the volume of the solid enclosed between the surfaces x = y 2 + z 2 and x = 1 − y 2 . (5 pts) x ey ˆ e x y ~ IV. Given F (x, y) = e ln y − i+ − e ln x ˆj, where x and y are positive. x y 1. Show that F~ is conservative .

(2 pts)

2. Find all potential functions of F~ . Z 3. Evaluate F~ • dr where C is any path from (1, 1) to (3, 3).

(5 pts) (3 pts)

C

V. Find the flux of F~ (x, y, z) = h3x, 3y, 6zi across the portion of the paraboloid z = 4 − x2 − y 2 given the outward orientation above the xy-plane. (4 pts) VI. Determine whether the given series is convergent or divergent.

1.

∞ X n=1

2.

cos

(−1)n 1 + n n

3.

(2 pts)

∞ X n=1

∞ X 2n2 − 3 (n + 3)10 n=1

(n2

1 + 1) tan−1 n

(4 pts)

(3 pts)

VII.Determine the interval of convergence of VIII. Consider the function f (x) =

√ 3

∞ X (x + 4)n n22n n=1

(5 pts)

x

th

1. Find the 4 degree Taylor polynomial for f centered about a = 27. (You do not need to give a general formula for the coefficients of the power series.) (3 pts) √ 3 2. Approximate 28 using the polynomial in (1). (2 pts) Bonus: (3 pts) If

∞ X

cn 4n is convergent, does it follow that the series

n=0

∞ X n=0

End Total: 50 Points

1

cn (−2)n is convergent? Explain.