1-10 /50 11 /30. MATH 251 Final Exam Fall 2015 Sections 512 Version A Solutions P. Yasskin 12 / 5 13 /20

Transcription

1 Name ID - /5 MATH 5 Final Exam Fall 5 Sections 5 Version A Solutions P. Yasskin Multiple hoice: (5 points each. No part credit.) / / 5 / Total /5. A triangle has vertices A,,, B,, and, 5,. Find the angle at A. a. orrect hoice b. 5 c. 6 d. e. 5 Solution: AB,, A,, cos AB A 9 AB A 6. Find the tangent plane to the graph of z x x at x,,. Where does it cross the z-axis? a. 7 b. 8 orrect hoice c. d. e. 8 Solution: fx, x x f, f x x, x x f x, 6 f x, x x f, 8 z f tan x, f, f x, x f, 6x 8 z-intercept f tan, 6 8 8

2 . Find the tangent plane to the graph of hperbolic paraboloid 9x z at the point,,. Where does it cross the z-axis? a. 9 b. 9 c. d. 9 orrect hoice e. 9 Solution: P,, F 9x z F 8x,8,8z N F P 8,8,8 N X N P 8x 8 8z z-intercept at x : 8z 76 z /. Sketch the region of integration for the integral tan then select its value here: dx d in problem (), a. b. c. d. e. orrect hoice Solution: everse the order of integration: / tan dx d dx dx x d dx x

3 5. Ham Duet is fling the Millenium Eagle through the galax on the path rt t, t, t. At t hours, he releases a trash pod which travels along the tangent line to the path of the Eagle with constant velocit equal to the velocit of the Eagle at time of release. Where is the trash pod hours after release? a., 8, orrect hoice b., 9, 7 c.,, 8 d.,, e. 5,, Solution: P r,, 8 v, t, t v,, The path of the trash pod is X P tv,, 8 t,, t, t, 8 t hour after release, the trash pod is at X,, 8, 8, 6. onsider the surface of the cone given in clindrical coordinates b z r above the x-plane. It ma be parametrized b r, rcos, rsin, r. Its temperature is T z. Find its average temperature. a. b. 6 c. d. orrect hoice e. 8 Solution: We first find the normal and its length: î k e r cos sin e rsin rcos The surface area is N e e î rcos rsin k rcos rsin rcos, rsin, r N r cos r sin r S rdrd r r 6 The integral of the temperature, T z r, is TdS r rdrd rr dr So the average temperature is r r 6 T ave A TdS 6 6 6

4 7. Death star is basicall a spherical shell with a hole cut out of one end, which we will take as centered at the south pole. In spherical coordinates, it fills the region between and. Find the volume. a. V b. V c. V 6 orrect hoice d. V 5 e. V 6 Solution: The volume is / V / sind dd cos cos cos 6 8. ompute f ds for f xe counterclockwise around the polar curve r cos shown at the right. Hint: Use a Theorem x a. e b. e c. e e d. e e e. orrect hoice Solution: B the FT, f ds fb fa because B A no matter what point ou start at.

5 9. ompute F ds for F x, x P counterclockwise around the complete boundar of the plus sign shown at the right. Hint: Use a Theorem x - - a. b. 6 orrect hoice c. d. e. 6 Solution: P x Q x x Q P B Green s Theorem, F ds x Q P dx d dx d area 6. ompute F ds over the complete surface of the hemisphere z 9 x V outward,for F x z, z, z Hint: Use a Theorem. oriented a. 7 b. 5 c. 8 d. 6 e. orrect hoice Solution: B Gauss Theorem F ds F dv. The divergence is H H F x z zz zx z cos and the volume element is dv sind dd. So the integral is: H F dv / cos sind dd sin /

6 Work Out: (Points indicated. Part credit possible. Show all work.). ( points) Verif Stokes Theorem F ds F ds for the vector field F z,xz, z and the surface of the cone z x for z, oriented down and out. Be careful with orientations. Use the following steps: Left Hand Side: The cone,, ma be parametrized as r, rcos, rsin, r a. ompute the tangent vectors: e r cos, sin, e rsin, rcos, b. ompute the normal vector: N îrcos rsin k rcos rsin rcos,rsin, r This is oriented up and in. Need down and out. everse: N rcos, rsin,r c. ompute the curl of the vector field F z,xz, z : î xz z k z z xz, z,z F x z î k z xz z d. Evaluate the curl of F on the cone: F r, r cos, r sin,r e. ompute the dot product of the curl of F and the normal N. F N r cos r sin r r f. ompute the left hand side: F ds F N drd r drd r 6

7 ight Hand Side: g. Parametrize the circle, : r cos, sin, h. Find the tangent vector on the curve: v sin, cos, This is counterclockwise. Need clockwise. everse v sin, cos, i. Evaluate F z,xz, z on the circle: F r 8 sin,8 cos, 8 j. ompute the dot product of F and the tangent vector v: F v 6 sin 6 cos 6 k. ompute the right hand side: F ds F vd 6 d which agrees with (f). 7

8 . (5 points) Sketch the region of integration for the integral / tan Shade in the region. dx d. You computed its value in problem () x. ( points) A cardboard box needs to hold 8 cm. The cardboard for the vertical sides costs per cm while the thicker bottom costs 6 per cm. There is no top. What are the length, width, height and cost of the box which costs the least? Solution: The volume constraint is V LWH 8. The cost is 6LWLHWH 6LWLHWH which needs to be minimized. Lagrange Multipliers: 6WH, 6LH, LW V WH, LH, LW V 6WH WH 6LWLH LWH 6LH LH 6LWWH LWH 6LWLH 6LWWH LH WH L W LW LW LHWH LWH 6LWWH LHWH 6LW LH W H LWH WW W 8 W 7 W L H 6LWLHWH 6 8