Calculus III 110.202 Final Exam. Johns Hopkins University, December 11, 2009.
Name: Signature: ... Not including this cover page, there are 17 pages in this
exam. The last ... your solutions and the problems. 2. This is a closed ... (1.5) If F
is a smooth vector field defined on the unit sphere S which has the outward
normal ...
Calculus III 110.202 Final Exam Johns Hopkins University, December 11, 2009
Name: Signature: Circle your section number:
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Instructions: 1. Not including this cover page, there are 17 pages in this exam. The last three pages are left blank intentionally. Feel free to write your solutions on the blank pages if necessary, but make sure to give directions to match your solutions and the problems. 2. This is a closed book closed notes exam. No calculators, no collaborations, no talking. Cheating is punished severely. 3. A correct answer but with no argumentations does not guarantee any points. An incorrect answer but with some correct steps always guarantees some partial credits. 4. If you do not understand a question or notation, ask the lecturer or TA. 1
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Problem 1. [8 × 5 = 40 points] True or false, no argument needed. (1.1) Let ~u and ~v be nonzero vectors in R3 . Then |~u + ~v | ≤ |~u| + |~v |. (1.2) Let ~u, ~v and w ~ be nonzero vectors in R3 . If ~u · (~v × w) ~ = |~u||~v ||w|, ~ then ~u ⊥ ~v , ~v ⊥ w ~ and ~u ⊥ w. ~ (1.3) An irrotational vector field on R3 − (0, 0, 0) is conservative. (1.4) Let f (x, y) be a real valued function. If both are continuous, then
2
∂ f ∂x∂y
=
∂2f ∂x∂y
and
∂2f ∂y∂x
exist and they
2
∂ f ∂y∂x .
(1.5) If F~ is a smooth vector field RR defined on the unit sphere S which has the ~ = 0. outward normal vector, then S Curl(F~ ) · dS (1.6) Every smooth surface has at least one orientation. (1.7) Let f = f (x, y) be a smooth function and p = (a, b) be a critical point 2 2 2 ∂2f of f . If ∂∂xf2 (p) < 0 and ∂∂xf2 (p) ∂∂yf2 (p) − [ ∂x∂y (p)]2 > 0, then f (p) is a local maximum value. (1.8) Let f (x, y, z) be a function. If the iterated integral Z z1 Z y1 Z x1 f (x, y, z) dxdydz z0
y0
exists, then the triple integral
x0
RRR
f (x, y, z) dV exists. Here V is the rect-
V
angular box [x0 , x1 ] × [y0 , y1 ] × [z0 , z1 ].
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Problem 2.[5+5 = 10 points] Let P = (0, 0, 1), Q = (1, 0, 3) and R = (−1, 2, 1) be three points in R3 . (2.1) Find the area of the triangle which has P, Q and R as the three vertices.
(2.2) Find the distance from the origin to the plane which contains P, Q and R.
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3.[5 + 5 + 7 + 8 = 25 points] (3.1) Find the tangent plane of the graph of x2 + y 2 − z 2 = 1 at (1, 1, 1).
− (3.2) Find the directional derivative at (0, 0, 1) along the vector → v = h−1, 2, 0i 2 x2 +y of the function f (x, y, z) = z − e .
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(3.3) Find a parametric equation of the tangent line at (1, 0, 1) to the following curve. x2 + y 2 = 1, z = x2 .
(3.4) Let f (x, y, z) be a smooth function and P (t) = (x(t), y(t), z(t)) be a smooth path. If P (0) is a critical point of f (x, y, z), find the derivative of f ◦ P at t = 0.
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4. [15 points] Find the absolute minimum value and maximum value, if they exist, of the function f (x1 , x2 , · · · , xn ) = (1 + x1 )(1 + x2 ) · · · (1 + xn ) with the constraint x1 x2 · · · xn = 2n . We require that x1 ≥ 0, x2 ≥ 0, · · · , xn ≥ 0.
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5.[10 points] Evaluate the integral
R4R2 0
√
6
y
cos (1 + x3 )dxdy.
6.[15 points] Find the area of the domain D which is in the first quadrant and bounded by xy = 1, xy = 3, y = x, y = 3x.
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7.[10+10 = 20 points] Let V be the solid bounded from below by z = and bounded from above by x2 + y 2 + z 2 = 2. (7.1) Find the volume of V .
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p
x2 + y 2
(7.2) Find the area of the boundary of V .
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8.[15 points] For each of the following vector fields, if it has a potential function, find one. If it does not have a potential function, explain why. F~ = hyz, xz, xyi
~ = hsin(yz), sin(xz), sin(xy)i G
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9.[15 points] Let C be the oriented upper half unit circle with initial point (−1, 0) and terminal point (1, 0). Let P = x + yexy + sin(x) and Q = x + xexy + cos(y). R Evaluate the path integral C P dx + Qdy.
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10.[15 points] Let S be the upper RR half unit sphere with outward normal vectors. ~ where Compute the surface integral S F~ · dS F~ = hx + xyeyz , xyexz , 1 − exz − eyz i.
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y x z 11.[10 + 10 = 20 points] Let F~ = h1 − x2 +y 2 , 1 + x2 +y 2 , e i be a vector field. R (11.1) Evaluate the path integral S F~ · d~s where S is the curve
x 2 + y 2 = a2
and z = 0
where a is some positive number. Assume that S is counterclockwise oriented if viewed from the +∞ of z-axis.
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(11.2) Evaluate the path integral
R C
F~ · d~s where C is the curve
z = x2 + y 2 − 4 and x + y + z = 100. C is counterclockwise oriented if viewed from the +∞ of z-axis.
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(Extra space)
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