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.