FINAL FOR CALCULUS
Time: 8:10–10:00 AM, Friday, June 22, 2001 Instructor: Shu-Yen Pan
No calculator is allowed. No credit will be given for an answer without reasoning.
1. [10%] Let F(x, y) = x3y4i + x4y3j. Find a function f such that ∇f = F and compute the line integral R
CF · dr where C is the curve r(t) =√
ti + (1 + t3)j, 0 ≤ t ≤ 1.
2. [10%] Use Stoke’s theorem to compute the integralRR
ScurlF · n dS where F(x, y, z) = yzi + xzj + xyk and S the part of the sphere x2+ y2+ z2 = 4 that lies inside the cylinder x2+ y2 = 1 and above the xy-plane.
3. [10%] Find the flux of the vector field F(x, y, z) = xi + yj + y2k over the unit sphere x2+ y2+ z2= 1.
4. [10%] Use Green’s theorem to find the area enclosed by the ellipse x2
a2 +y2 b2 = 1 for some a, b > 0.
5. [10%] Sketch the solid whose volume is given by the iterated integral Z 2
0
Z 2−x
0
Z 4−x2
0
dz dy dx.
6. [10%] Find local maximum and minimum values and saddle points of the function f (x, y) = x2+ y2+ x2y + 4.
7.
(i) Use implicit differentiation to find ∂z/∂x for xy2+ yz2+ zx2= 3.
(ii) Find a unit normal vector of the plane passing through the point (0, 0, 1) and spanned by the two vectors i + j and j − 2k.
8. [10%] Find the length of the curve r(t) = t2i + 2tj + ln tk for 1 ≤ t ≤ e.
9. [10%] Test the series
X∞
n=1
cos nπ
√n
for absolutely convergence, conditionally convergence or divergence.
10. [10%] Let f (x) = exx−1.
(1) Find the power series representation of f in powers of x.
(2) Differentiate the power series in (1) and show that X∞
n=1
n
(n + 1)! = 1.
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