1. (7 points) Let f(, ) = +3 + +. Find and classif each critical point of f as a local minimum, a local maimum, or a saddle point. Solution: f = + 3 f = 3 + + 1 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
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.
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
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) 1 1 1 3 4 1 (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
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. 9. 5 3 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. 15. 6π
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 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/. 1 13. (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
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 = +. 15. (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 = 8 16. (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. 1 5. 1 1
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 = 1 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 1 1 1 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