ASU MAT 267 - mat_267_-sp14-test_2_reviews (6 pages)

TEST 2 REVIEWS 11 3 Partial Derivatives 1 Find f x for f x y 3x5 2 x3 y 2 9 xy 4 2 Find the first partial derivatives of the function of c ln a a 2 b2 3 Find 2 z for z y tan8x x 2 4 Find f x for f x y yx cos t8 dt 5 Find all the second partial derivatives of f x y 4 x3 y 7 xy 2 11 4 Tangent Planes Linear Approximations 1 Find the linearization L x y of the function at the given point Round the answers to the nearest hundredth f x y x y 5 4 2 Find the differential of the function u e5t sin 3x 3 Find the equation of the tangent plane and the normal line to the given surface at the specified point f x y 4 x3 y 7 xy 2 1 1 3 4 Find an equation of the tangent plane to the given surface at the specified point z 18 x 2 2 y 2 3 2 1 5 Use differentials to estimate the amount of metal in a closed cylindrical can that is 13 cm high and 6 cm in diameter if the metal in the top and bottom is 0 09 cm thick and the metal in the sides is 0 01 cm thick rounded to the nearest hundredth 11 5 The Chain Rule w 1 Use the Chain Rule to find where s 3 t 0 s w x2 y 2 z 2 2 Use the Chain Rule to find x st y s cos t z s sin t u p u x p 6r 5t x y y z y p 6r 5t z p 6r 5t 11 6 Directional Derivatives The gradient vector 1 Find the direction in which the maximum rate of change of f at the given point occurs f x y sin xy 1 0 2 Find the gradient of the function f x y z x 2 e y z 3 If f x y x 2 9 y 2 use the gradient vector f 10 2 to find the tangent line to the level curve f x y 136 at the point 10 2 4 Find the equation of the tangent plane to the given surface at the specified point 5x2 3 y 2 8z 2 353 3 6 5 5 The temperature of a gas at the point x y z is given by G x y z x2 5xy y2z a What is the rate of change in the temperature at the point 3 2 1 in the direction v 2 1 4 b What is the direction of the maximum rate of change of temperature at the point 3 2 1 c What is the maximum rate of change at the point 3 2 1 6 Find the directional derivative of the function at the given point in the direction of the vector v y f x y z x tan 1 8 8 8 v 10i 7 j 7k z 11 7 Maximum minimum values 1 Find the critical points of the function y4 f x y 5 76 xy 38 x 240 y 4 2 2 Find the local maximum and minimum value and saddle points of the function f x y x2 xy y 2 9 x 6 y 10 3 Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is 24 4 Find the minimum of the function 2 f x y x2 2 y 2 2 xy 2 x 3 y subject to x y 1 5 Suppose 1 1 is a critical point of a function f with continuous second derivatives In the case of f xx 1 1 7 f xy 1 1 8 f yy 1 1 10 what can you say about f 6 Find all the saddle points of the function f x y x sin y 3 7 Find the absolute maximum value of the function f on the set D f x y 3x 2 8 y 2 10 x 2 y 9 D x y x 1 y 1 12 1 Double Integrals over Rectangles 1 Calculate the double integral 2 Calculate the iterated integral 0 x sin x y dA R 0 6 2 R 6 3 0 0 x y dxdy ln 4 ln 5 5 x y 3 Calculate the iterated integral dxdy e 0 0 4 Calculate the double integral 2 3 4 3x y 5 x dA R x y 0 x 1 0 y 4 R 5 Calculate the double integral xy 2 dA R x y 0 x 2 1 y 1 2 R x 4 6 Calculate the double integral y xye dA R x y 0 x 2 0 y 1 R 12 2 Double Integrals Over general regions 1 Find the volume of the solid in the first octant bounded by the cylinder z 9 y 2 and the plane x 1 2 Evaluate the iterated integral 5 5 1 y xy dxdy 3 Evaluate x 2 y 2 dA where D is the figure bounded by y 1 y 2 x 0 and x y D 4 Evaluate the double integral y 3 dA where D is the triangular region with vertices D 0 1 7 0 and 1 1 5 Evaluate the integral by reversing the order of integration 1 4 0 4 Compute 2 e x dxdy 4y 4 x 2 y 2 dA where D is the disk x 2 y 2 4 by first identifying the D integral as the volume of a solid 12 3 Double Integrals in Polar coordinates 1 Evaluate the integral by changing to polar coordinates e x 2 y2 dA where D is the D region bounded by the semicircle x 4 y 2 and the y axis 2 Use polar coordinates to find the volume of the solid inside the cylinder x 2 y 2 9 and the ellipsoid 2 x 2 2 y 2 z 2 36 3 Use polar coordinates to find the volume of the solid under the paraboloid z x 2 y 2 2 2 and above the disk x y 9 4 Use polar coordinates to find the volume of the solid bounded by the paraboloid z 7 6 x 2 6 y 2 and the plane z 1 5 Evaluate the iterated integral by converting to polar coordinates 2 2 4 y 2 0 x 2 y 2 3 2 dxdy ANSWERS 11 3 Partial Derivatives 1 15x4 6 x2 y 2 9 y 4 2 1 a b 2 2 b a a b a 2 b2 2 2 3 128 y sec2 8x tan 8x 4 cos x8 5 f xy 12 x 2 14 y f yx 12 x 2 14 y f xx 24 xy f yy 14 x 11 4 Tangent Planes Linear Approximations Differentials 1 L 2x 1 25 y 5 2 du 5e5t sin 3x dt 3e5t cos 3x dx 3 5x 10 y …