D = 2(2) 3 2 = 4 9 = 5 < 0
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1 1. (7 points) Let f(, ) = Find and classif each critical point of f as a local minimum, a local maimum, or a saddle point. Solution: f = + 3 f = f = f = 3 f = Both f = and f = onl at ( 3/5, /5). Using the second derivative test we have = () 3 = 4 9 = 5 < So therefore ( 3/5, /5) is a saddle point.. (7 points) Epress the volume of the solid that lies within both the clinder + = 1 and the sphere + + z = 4 as an integral using clindrical coordinates and evaluate it. Solution: The sphere can be epressed as z = ± 4 = ± 4 r. So the volume can be epressed in clindrical coordinates as: V = 1 dv = π 1 4 r = π 1 4 r r dz dr dθ r 4 r dr = 4π [ 1 (4 3 r ) 3/] 1 = 4π [ ] 8 3 3/ 3
2 3. Let F(, ) = zi + zj + ( + z)k and answer the following questions (a) (7 points) Find a function f so that f = F. Solution: One possible answer is f = z + z. Can check that f = z f = z f z = + z (b) (4 points) Let C be a curve given b vector equation C : r(t) = cos(t)i + sin(t)j + t k, t [, π/]. π Evaluate the integral z d + z d + ( + z) dz. C Solution: r() = 1,, and r(π/) =, 1, 1 so b the fundamental theorem of line integrals z d + z d + ( + z) dz = f(, 1, 1) f(1,, ) C = 1 = 1 (c) (3 points) Let line segment C 1 in R 3 go from P (1,, ) to R(1, 1, 1), and line segment C go from R to Q(, 1, 1). Evaluate the integral. C 1 C z d + z d + ( + z) dz. Solution: Because the paths start and end at the same points as (b) we know conservative vector field have path independent line integrals so the answer must also be 1.
3 4. (7 points) The solid lies above the cone z = + and below the sphere + + z = 4. Use spherical coordinates to evaluate the integral z dv of the height function z over the solid. Solution: The sphere can be epressed as ρ = 4 = ρ =. The cone can be epressed as z = r z = r ρ cos φ = ρ sin φ cos φ = sin φ φ = π/4 So z dv = π π/4 ] π/4 [ [ sin φ ρ 4 = π 4 [ ] 1 = π [4] = π 4 (ρ cos φ)(ρ sin φ) dρ dφ dθ ] 5. (7 points) Use Green s theorem to find the work done b the force F = ( 3)i + ( )j on a particle moving counter-clockwise around the circle ( ) + = 4. Solution: W ork = F T ds = ( ) ( 3) da C = 1 ( 3) da = 1 da (area of the circle) = (π ) = 8π
4 6. Consider the integral π π sin d d (a) (3 points) Sketch the region of integration. Label all relevant intersection points. 4 π 3 (π, π) (b) (5 points) Evaluate the integral above b reversing the order of integration. Solution: π π sin d d = π π sin = sin d = cos() = π d d 7. (6 points) Find the surface area of the part of the half-cone z = + bounded from above b the plane z = 1. Solution: z = + z = + So surface area is given b (z ) + (z ) + 1 da = ( + ) + ( + ) da = da = da The cone is over the circle of radius 1 centered at the origin in the -plane. So using the area of a circle formula we have the final result = (π1 ) = π
5 Multiple Choice. Circle the best answer. No work needed. No partial credit available. 8. (4 points) The surface ρ = cos φ can be described as a A. Plane B. Half-Cone C. Sphere. Paraboloid 9. (4 points) Evaluate E A. 9 B. 7 C dv, where E = {(,, z) 3,, z + }. 1. (4 points) Find the mass of a wire that lies along the curve r(t) = (t 1)j + 3tk, t, if the densit is δ = 4z. A. 5 B. 115 C π
6 11. (4 points) Consider the function f(, ) = + + at the point ( 1, 1). In what direction does f decrease most rapidl? A. 1 1, 1 B. 1 1, 1 C. 1 1, , 1 1. (4 points) Find the work done b the force F =,, z over the curve r(t) = ti+t j+tk, t 1 in the direction of increasing t. A. 1/4 B. 1/3 C. 1/ (4 points) The vector field below could have been generated b which of the following? A. i + j B. i j C. i j. i j E. i ( )j
7 14. (4 points) Let F = +, + z,. Which of the following is true? A. curl F = + and div F = z +. B. curl F =,, and div F =. C. curl F =,, and div F =.. curl F = z,, + and div F = (4 points) Find the absolute maimum and minimum values of f(, ) = + on the set = {(, ) + 4}. A. min = 1, ma = 1 B. min = 4, ma = 4 C. min = 1, ma = 8. min =, ma = (4 points) Estimate the change of the function f(,, z) = ln( + + z ) if the point P (,, z) moves from P (1, 1, ) a distance of ds = 1 units in the direction of 3i + 6j k. 5 A. 1 B. 1 5 C
8 More Challenging Problem(s). Show all work to receive credit. 17. (6 points) Find the average value of the function f() = 1 cos(t ) dt on the interval [, 1]. Solution: For 1-variable functions over intervals (calc 1) we have the average value of a function is f ave = 1 1 = f d cos(t ) dt d There is no closed form integral of cos(t ) so we need to be clever. Through drawing a picture of the region and switching the bounds of integration = = 1 t 1 [ sin(t ) = cos(t ) d dt t cos(t )dt ] 1 = sin 1 (now use u-sub) 18. TRUE or FALSE? Circle the right choice. No work needed (a) ( points) If F and G are vector fields, then curl(f + G) = curl(f) + curl(g). A. TRUE B. FALSE (b) ( points) If F and G are vector fields, then curl(f G) = curl(f) curl(g). (c) ( points) A. TRUE B. FALSE A. TRUE B. FALSE (d) ( points) The integral e + sin d d = π z = + and the plane z =. r dz dr dθ = represents the volume enclosed b the cone A. TRUE B. FALSE
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