I. Let f (x, y) = x3 + y 2 − 3x + 2y. 1. Find the directional derivative of f at O(0, 0) along ~v = h12, 5i. 2. Determine what occurs at each critical point of f using the Second Derivative Test. II. Find the minimum and maximum values of f (x, y) = x2 + y along x2 + y 2 = 1. III. Set-up an iterated triple integral that yields the volume of the solid in the first octant above p p z = x2 + y 2 and below z = 4 − x2 − y 2 using: 1. rectangular coordinates 2. spherical coordinates IV. Let F~ (x, y) = hcos y − ye−x , e−x − x sin y + 1i. 1. Show that F~ is conservative without looking for a potential function. R ~ along any path C from (−π, 0) to (0, π). 2. Use FTLI to evaluate F~ · dR C
V. Consider the force field F~ (x, y) = h−y 2 , xyi. 1. Compute the work done by F~ in moving an object along ht3 + 1, 2ti from the point (1, 0) to the point (2, 2). 2. Let C be the positively oriented boundary of the region enclosed by the x-axis and the √ H ~ semicircle y = 4 − x2 . Use Green’s Theorem to evaluate F~ · dR. C
VI. Given F~ (x, y, z) = h3x, z, 9yi. 1. Obtain divF~ and curlF~ . 2. Evaluate the flux of F across the positively oriented portion of the plane 2x + 3y + z = 6 in the first octant. VII. Determine whether the series converges or diverges. 1.
∞ X 2n + 5 3n5/2 − 1 n=1
2.
∞ X e1/n n=1
3.
n2
VIII. Find the radius and interval of convergence of the power series
∞ X (x + 2)n
1 , where |x| < 3. (3 + x)2
X. Find the 3rd degree Taylor polynomial of
1 about 1. x
lnn n
n=1
n=1
IX. Obtain a power series representation for
∞ X (−1)n n2n
(n2 )(2n )
.