Math 114 Spring 2013 Final Exam

Transcription

1 1. Assume the acceleration of gravity is 10 m/sec downwards. A cannonball is fired at ground level. If the cannon ball rises to a height of 80 meters and travels a distance of 0 meters before it hits the ground, what is the magnitude of the initial velocity in meters per second? (A) 6 (B) 8 (C) 50 (D) 5 60 (F) 6 (G) 7 (H) 80. Find the equation of the plane that passes through ( 1,, ) and contains the line x y z = 1 + t = 1 t = + t The y-coordinate of the point where this plane intersects the y axis is (A) -1 (B) 0 (C) 1 (D) (F) (G) 5 (H) 6

2 r, ln cos,0 at t =.. Find the curvature for ( t) = t ( t) (A) 1 (B) (C) (D) (F) (G) (H). Find the arclength of the vector function / ( t ) = cos t + sin t + t r i j k for 0 t. (A) 1 (B) 1 (C) 16 (D) 18 0 (F) (G) 8 (H) 1

3 5. Let ( ) cos sin r t = t i + t j + t k Using the parametric equations for the line tangent to the function at t =, find the coordinates of the point where the tangent line intersects the xy plane. (A) ( 1,1,0 ) (B) ( 1, 1,0 ) (C) 1,1 +,0 (D) 1 +,1,0 1, + 1, 0 (F) 1,1, (G) ( 0,0,0 ) (H) The line does not intersect the xy plane. 6. Let and, 7. z = x y + x x = s + t y = s t Find z t when s = and t = 1. (A) (B) 7 (C) 8 0 (D) 5 (F) (G) 9 1 (H)

4 7. Let ( ) ( ) f x y z x y z,, = ln +. Using the linearization of f at ( 1,1,1 ), estimate the value of f ( ) (B) ln ( ) (C) ln ( ) 1. (D) ln ( ) + 1. ln ( ) (F) ln ( ) 1.6 (G) 0. (H) ,1.,1.1. f x, y = x xy + y. Find the local minimum of f. 8. Let ( ) (A) (B) 5 (C) 7 (D) (F) 7 (G) 16 (H)

5 9. Find the product of the maximum and minimum values of (,, ) ( ) ( 1) ( ) f x y z = x + y + z + on the sphere (B) 1 (C) 8 (D) 16 1 (F) 6 (G) 80 (H) 85 x + y + z = Compute the integral 1 x 0 0 ( x y) x + y dydx HINT: A change of variable might help (B) 1 (C) 9 (D) (F) 1 5 (G) (H)

6 11. Find the work done by the force field F = x i y j+ k on a particle as it moves along the helix given by (A) (B) (C) (F) 6 (G) (H) ( ) cos sin r t = t i + t j+ t k (D) from the point ( 1,0,0 ) to ( 1,0, ). 1. Consider the planar region D drawn below whose boundary consists of the curves C C C C oriented as shown. Suppose that ( x, y), 1,, and, functions and their partial derivatives are continuous on D, and that F is a vector field whose component Q P F dr = 1, F dr = 5, F dr =, and da = 1. x y C C C D 1 Evaluate F d r? C Carefully justify your answer. (A) 1 (B) 0 (C) 1 (D) 5 (F) 7 (G) 9 (H) 11 5

7 1. A particle moves along the line segments from ( 0,0,0 ) to ( 1,0,0 ) to ( 1,5,1 ) to ( 0,5,1 ) and back to ( 0,0,0 ) under the influence of the vector field ( x, y, z) = z + xy + y F i j k. Find the work done. (B) 1 (C) 7 (D) (F) (G) 81 (H) Let S be the portion of the surface z = xy lying inside the cylinder surface area S. (B) (C) (D) ( 1) (F) ( 1) (G) ( 1) (H) ( 1) x + y = 1. Compute the 6

8 15. A sphere of radius has a hole or radius 1 drilled straight through the center. What is the volume remaining? In terms of inequalities, the region is ( ) (A) (B) (C) 6 (D) ( ) ( 1 ) 9 (F) 10 (G) 5 (H) {,, and 1 } R = x y z x + y + z x + y. 16. Let (,, ) arctan ( ) ln ( ) F x y z = z y i + z x + j + z k. Find the outward flux of Fthrough S, the part of the paraboloid plane z = 5and is oriented upward. (B) (C) 6 (D) 8 16 (F) (G) 8 (H) x y z + + = 9that lies above the 7