Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.

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1 . Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function f(x, y, z) in cylindrical coordinates over the solid shown. Let a = and b =. a. f max =, f min = b. f max =, f min = c. f max =, f min = d. f max =, f min =. Find the directional derivative of f at the given point in the direction indicated by the angle. f ( x, y ) = x e y, ( 9, ), = a. 7 b. c. d. 9 f (rsin, rcos, z)r dz dr d f (rcos, rsin, z)r dz dr d f. g. 9 h. 7. valuate the double integral. f (rcos, rsin, z)r dz dr d f (rsin, rcos, z)r dz dr d x y da, = { x, x y x }. valuate the integral by changing to polar coordinates. xy da where is the disk with center the origin and radius. f. f (rsin, rcos, z)r dz dr d g. h. f (rsin, rcos, z)r dz dr d f (rcos, rsin, z)r dz dr d f (rcos, rsin, z)r dz dr d a. b. c. d. f.. valuate the integral by changing to polar coordinates. where R x + y da R = {(x, y) 9 x + y 8, y }. 7. Find the saddle point of the function f(x, y) = x y 9x y 8. Find the linearization L(x, y) of the function at the given point. f (x, y) = sin(x + y), (, ) 9. Find the differential of the function. u = e t sin 8x w. Use the Chain Rule to find where s =, t =. s w = xy + yz + zx x = st, y = e st, z = t f Find a vector function that represents the curve of intersection of the two surfaces: the circular cylinder x + y = and the parabolic cylinder z = x. PAG

2 7. valuate the integral. r(t) = cos(t) i + sin(t) j + cos (t) k r(t) = cos(t) i + sin(t) j + cos (t) k x da, r(t) = cos(t) i + sin(t) j cos (t) k r(t) = cos(t) i sin(t) j cos where is shown on the illustration below with a =. (t) k. At what point is the following function a local minimum? f(x, y) = 7x + 7y + 7x y + a. (, 7) b. (, ) c. (, ) d. (, ). valuate the double integral. x cos y da, where is bounded by y =, y = x, x = 7 a. cos 9 cos 9 b. cos 9 cos 9. valuate the double integral. (7x y) da, cos 9 f. cos 9 where is bounded by the circle with center the origin and radius 7. a. b. c.. d. a. b. c. d Find the area of the part of hyperbolic paraboloid z = y x that lies between the cylinders x + y = 9 and x + y =. 9 9 c Find the area of the part of paraboloid x = y + z that lies inside the cylinder z + y = 9. d. 7. Find the volume bounded by the paraboloid z = x + y + and the planes x =, y =, z =, x + y =.. valuate the integral by reversing the order of integration e x dxdy y nter your answer in terms of PAG

3 . valuate the triple integral. x y dv where lies under the plane z = + x + y and above the region in the xy plane bounded by the curves y = x, y =, and x =.. Set up, but do not evaluate, an integral expression for the moment of inertia about the where is bounded by the paraboloid x = y + z and the plane x = ; (x, y, z) = x + y + z. The choices are rounded to the nearest tenth. a. 9. b. 9. c. 87. d Use a triple integral to find the volume of the solid bounded by the cylinder x = y and the planes z = and x + z =. y (x + y y a. ) dy dz dx b. y y + z y y + y (x + y ) (x + y + z y ) dx dz dy (x + y y + z y y + z The choices are rounded to the nearest tenth. a..9 b. y (x + y ) (x + y + z ) dx dz dy y + z c.. d. 8.. xpress the integral f (x, y, z) dv as an iterated integral of the form b v ( x ) d ( x,y ) f dz dy dx, a u ( x ) c ( x, y ) solid bounded by the surfaces x + z =, y =, and y =. where is the. valuate y ds, where C is given by x = t, y = t, t. C. valuate the line integral F dr, where F(x, y) = x y i y x j C and C is given by r(t) = t i t j, t. a.. b.. c.. d..8. z f (x, y, z) dz dy dx x f (x, y, z) dz dy dx x. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x f (x, y, z) dz dy dx x f (x, y, z) dz dy dx x = ln t, y = t, z = t 7 ; (,, ). x = + t, y = t, z = + 7t x = t, y = + t, z = + 7t x f (x, y, z) dz dy dx x c. x = t, y = + t, z = + t d. x = t, y = t, z = 7t x = t, y = + t, z = + t 7. Find r(t) if r (t) = t i + t j t k and r() = j. PAG

4 8. A contour map for a function f is shown. Use it to estimate the value of f (, ).. Which plot illustrates the vector field F x, y? ( ) = x y, x + y I. II. a. b. c. d. III. IV. 9. Find equation of the tangent plane to the given surface at the specified point. 7 x + y + z = 7, (,, 7 ). Find the limit, if it exists lim (x,y) (,) xy x + y. Use spherical coordinates to evaluate a. b. x + y + z dv c. The limit does not exist z. Use quation 7 to find. x 8xy + yz + zx = 7. A particle moves with position function r(t) = ( t t 9) i + t j Find the tangential component of the acceleration vector. where is bounded below by the cone sphere =. = and above by the The choices are rounded to the nearest hundredth. a..9 b..9 c.. d..9. f. 8. a. a T = t b. a T = t c. a T = t d. a T = t + 9 PAG

5 ANSWR KY. b. h.. a. d. c (,) L=.x.y du=e t sin( 8x) dt+8e t cos( 8x) dx. 8. a. c. b. b. + ( e 9 ). 7. d 8. b 9. e. c. e. e. d..7. b. b t i+ ( t +) j t 8. b 9. 8x y+z=7. b z x = 8y +x z. 8y z+x. b. I. c k PAG