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Guidelines for the test:
• Put your name or student ID number on every page.
• If you got more than 100 points, you will only get 100 points. (超過100分以100分計 算)
• The exam is closed book; calculators are not allowed.
• Please show all work, unless instructed otherwise. Partial credit will be given only for work shown. Print as legibly as possible - correct answers may have points taken off, if they’re illegible.
• 第八題及第九題只需要寫出完整積分式, 不需要把體積算出來.
• Mark the final answer.
1. (5 pts) Suppose thatg(s, t) = f(x(s, t), y(s, t)), wheref is a differentiable function of xand yand where x=x(s, t) and y=y(s, t) both have continuous first-order partial derivatives. Use the table of values to calculate gs(0,0).
f g fx fy x y xs xt ys yt
(0,0) 3 6 4 8 1 2 1 1 1 0
(1,2) 6 3 2 5 3 6 3 6 4 8
2. (10 pts) Find the critical points of the given function and determine the type of the critical points (local maximum, local minimum, or saddle)
f(x, y) =x3−3xy+y3
3. (10 pts) Use Lagrange Multipliers to find the maximum and minimum values of the function f(x, y) =exy subject to the constraint x2+y2 = 8
4. (5 pts) Given f(x, y) =
∫ x y
cos (t2)dt, find fx and fy.
5. (5 pts) Evaluate the iterated integral by first changing the order of integration.∫ 1
0
∫ 1
y
ex2dx dy
6. (5 pts) Evaluate
∫ 1 0
∫ 1 0
emax (x2,y2)dxdy
7. (10 points) Evaluate the integral
∫ ∫
R
y+ 3x dA, where R is the region bounded by y = 3−3x, y= 1−3x, y=x−3 and y=x−1, by making an appropriate change of variables.
8. (5 pts) Set up but Do Not Evaluate the triple integral for the volume of the tetrahedron enclosed by the coordinate planes and the plane 2x+y+z = 4
9. (10 pts) Q is the solid that lies above the cone z = √
x2+y2 and below the sphere x2+y2+z2 =z.
(a) Set up but Do Not Evaluate the triple iterated integral for the volume of Q in Cylindrical coordiantes.
(b) Set up but Do Not Evaluate the triple iterated integral for the volume of Q in Spherical coordiantes.
10. (a) (5 pts) Given that F⃗ =< y, x >, find an f(x, y) such that F⃗ =∇f
(b) (5 pts)∫ 設路徑r為由點(1,0)沿橢圓 x122 + y222 = 1 之上半橢圓到(−1,0), 則線積分
r
(y dx+x dy) =?
11. (5 pts) Evaluate the line integral
∫
C
xsinz ds, whereC is the circular helix given by the equations x= cost,y = sint, z =t, 0≤t ≤2π.
12. (a) (5 pts) Given F⃗ =< xysinz,cosxz, ycosz >, compute div ⃗F
(b) (5 pts) Use the Divergence Theorem to calculate the surface integral∫ ∫
SF⃗·d ⃗S, whereSis the ellipsoidx2/12+y2/22+z2/32 = 1 andF⃗ =< xysinz,cosxz, ycosz >
13. Given F⃗ =< x2, y4 −x, z2sinz+x >, (a) (5 pts) compute curl ⃗F,
(b) (5 pts) use Stokes’ Theorem to evaluate∫
CF⃗·d⃗r, whereCis the circlex2+y2 = 9 on thexy-plane, oriented so that it is traversed counterclockwise when view from the positive z-axis. (C 為一圓且由上方看為逆時針運動)
