Guidelines for the test:
• Put your name or student ID number on every page.
• There are 12 problems: 6 problems in Part I (7 points each) and 6 problems in Part II . If you got more than 100 points, only 100 points were counted. (超過 100分以100分計算)
• The exam is closed book; calculators are not allowed.
• There is no partial credit for problems in the Part I (multiple-choice (選擇) and fill-in (填充) problems).
• For problems in the Part II (problem-solving (計算題) problems), please show all work, unless instructed otherwise. Partial credit will be given only for work shown. Write as legibly as possible - correct answers may have points taken off, if they’re illegible.
• Mark the final answer.
Part I: (7 points for each problem)
Multiple Choice - Single Answer (選擇題- 單選題).
(1) Given f (x, y) = x3− 3x + y2+ 9, which of the following is correct?
A) (-1,1) is a saddle point B) (2,0) is a local minimum C) (1,0) is a local minimum D) all of the above
(2) R is a region bounded by y = 4x + 2, y = 4x + 5, y = 3− 2x and y = 1 − 2x, Given the transformation u = y − 4x,v = 2x + y, which of the following is equivalent to ´ ´
Rx2dA?
A) ´5
2
´3
1 x2dudv, B) ´5
2
´3
1((v−u)6 )2dudv, C)´5
2
´3
1((v−u)6 )2(−13) dudv D) ´5
2
´3
1((v−u)6 )2 13dudv E) None of the above. Ans=?
Fill-In Problems(填充)
(3) Find an equation of the plane containing the point (3, 2, 1) with normal vector
< 4, 5, 6 >.
Ans = .
(4) Given that f (x, y) = x3− 3x + y2+ 9, compute the directional derivative of f at (1, 2) in the direction of the vector <
√3 2 ,12 >
Ans = .
(5) Suppose that g(t) = f (x(t), y(t)), where f is a differentiable function of x and y and where x = x(t) and y = y(t) both have first-order derivatives. Given that
x0(1) = 2, y0(1) = 3 x(1) = 4, y(1) = 5 , f (4, 5) = 6, fx(4, 5) = 7, fy(4, 5) = 8,
g0(1) = .
(6) Find the equation of the tangent plane to the surface at the given point.
z = x2− y2+ 1 at (1, 2,−2)
Ans = .
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Part II:
Problem-Solving Problems (計算題 Show all work) (7) Given that r(t) =< t2− t, e3t, 0 >, calculate
• limt→0r(t) =?
• dtdr(t) =?
• ´
r(t) dt =?
• ´1
0 r(t) dt =?
(8) Show that the limit does not exist.
lim
(x,y)→(0,0)
12xy2 x2+ y4
(9) Given f (x, y) = x3− exy + y3, find the partial derivatives, fx, fy, fxy and fxx.
(10) For a differentiable function g(u, v) = f (x(u, v), y(u, v)) with f (x, y) = 10x1/4y3/4, x(u, v) = u + v and y(u, v) = u− v, use the chain rule to find gu(u, v) and gv(u, v).
Page 3
(11) Find the maximum and minimum values of the function f (x, y) = x2+ y2+ 6y subject to the constraint x2+ 4y2≤ 4
(12) Evaluate the iterated integral by first changing the order of integration.
ˆ 2
0
ˆ 1
y/2
ex2dx dy
