Final Exam Math 222 Spring 2011 May 11, 2011 Name: Recitation Instructor s Initials: You may not use any type of calculator whatsoever. (Cell phones off and away!) You are not allowed to have any other notes, and the test is closed book. Use the backs of pages for scrapwork, and if you write anything on the back of a page which you want to be graded, then you should indicate that fact (on the front). Except for the last page which is the cheat sheet (and which you should not hand in) do not unstaple or remove pages from the exam. By taking this exam you are agreeing to abide by KSU s Academic Integrity Policy. Simple or standard simplifications should be made. Box your final answers when it is reasonable. You must show your work for every problem, and in order to get credit or partial credit, your work must make sense! MAY THE FORCE BE WITH YOU, YOUNG JEDIS!!! Problem Possible Score Problem Possible Score 1 10 6 15 2 15 7 20 3 10 8 15 4 10 9 20 5 20 10 15 Total 65 85 Cheat Sheet Bonus: cos 2 θ = 1 + cos(2θ) 2, sin 2 θ = 1 cos(2θ), sin(2θ) = 2 sin θ cos θ. 2
1. The first question refers to the following figure. It is drawn to scale and u is perpendicular to r. r p w v The vectors shown are all of unit length. They all lie in the same plane. u List the following values in order from most negative to most positive: a) u v b) u r c) u r d).1 e) u u f) u p g) v w h) u w i) v w j) +1.1 2. Find the equation for the plane containing the points: (2, 2, 2), (1, 1, 0), and (0, 1, 1).
3. Let C be the curve given by ( r(t) = (cos t)(cos 2 2t)(cos 2 3t), (sin t)(sin 2 2t)(sin 3 3t), with 0 t π. Compute the following integral: (3x 2 e (x3 +y 2 +z), 2ye (x3 +y 2 +z), e (x3 +y 2 +z) ) d r. C ) t 2 π 2,
4. Let S be the set of points within the set: 0 x 4, 0 y 5, 0 z 2. If n is the outward unit normal to this region, then compute (in an intelligent way!): (2x 3y 2 z, y + 2e xz, 3z sin(x 2 y)) n da. S
5. A certain differentiable function satisfies: (a) f(9, 7) = 11, and f(2, 1) = 7. (b) f(9, 7) = (6, 8), and f(2, 1) = ( 5, 12). At each of the two points in question (i.e. answer the following questions: at (9, 7) and at (2, 1)) (a) In what direction is the function increasing the fastest and what is the rate of change in that direction? (b) What is the directional derivative in the direction of < 3, 4 >? (Note: just to be completely clear about semantics here, you are supposed to give the same directional derivative at each point. I did not ask for the directional derivative in the direction of the point (3, 4).) (c) What is the tangent plane and/or the linear approximation at each of the two points?
6. Find the maximum and minimum of the function on the set f(x, y) = x 2 2x + y 2 + 2y g(x, y) = x 2 + y 2 50. Show your work carefully, and explain what you are doing. (No essays, please. Just a few short words in the right places will suffice.)
7. Let E be the subset of x 2 + y 2 + z 2 = 16 which also satisfies Express 2 z 2, and x 0. E x 2 ds as an iterated integral (i.e. a double or triple integral) over a subset of IR 2 or IR 3. You do NOT need to find this integral.
8. Let T be the tetrahedral with vertices Express (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 1). T f(x, y, z) dv as one triple integral whose bounds of integration are simple linear functions. To save you time here, I will give you the following: The four planes which bound this tetrahedral are: x = 0, y = 0, x + y + z = 1, and x + y z = 1. Note: Not every order of integration works...
9. Let E be the part of the set x 2 + y 2 + z 2 9 that is contained in the set x 0, y 0. Find: 8x dv. E
10. Let S be the part of the cone z = x 2 + y 2 which is between the planes z = 0 and z = 4. Find the surface area of this cone.
Integral Definitions and Basic Formulas: Line Integrals: F ( r) d r = C b C a b f( r) ds = f( r) dx = a b C a F ( r(t)) r (t) dt, Orientation Matters! f( r(t)) r (t) dt, f(x(t), y(t), z(t)) x (t) dt, Orientation Doesn t matter! Orientation Matters! Surface Integrals: With N := r u r v 0, and with r(r) = S we have F n ds = F ( r(u, v)) N(u, v) du dv Orientation Matters! S f( r) ds = R S R f( r(u, v)) N(u, v) du dv Orientation Doesn t matter! Green s Theorem: If D is a region in the plane, and D has positive orientation (i.e. has counter-clockwise orientation), then ( Q P (x, y) dx + Q(x, y) dy = x P ) da. y D D Stokes s Theorem: S ( F ) n da = S F d r.
(If n is pointing right at you then orient S in a positive fashion (i.e. counter-clockwise fashion, typically) to make the identity hold.) Spherical Coordinates: x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ dv = ρ 2 sin ϕ dρ dϕ dθ. Second Derivative Test: Suppose the second partial derivatives of f are continuous on a disk with center (a, b) and suppose f(a, b) = (0, 0). Let D = D(a, b) = f xx (a, b)f yy (a, b) f xy (a, b) 2. (a) If D > 0 and f xx (a, b) > 0, then (a, b) is a local minimum. (b) If D > 0 and f xx (a, b) < 0, then (a, b) is a local maximum. (c) If D < 0, then (a, b) is a saddle point.