TEST 2 REVIEWS 11.3: Partial Derivatives 1. Find xffor 5 3 2 4( , ) 3 2 9f x y x x y xy   2. Find the first partial derivatives of the function of 22ln( )c a a b   3. Find 22zx for tan8z y x 4. Find xffor 8( , ) cos( )xf x y t dty 5. Find all the second partial derivatives of 32( , ) 4 7f x y x y xy 11.4: Tangent Planes & Linear Approximations 1. Find the linearization L(x, y) of the function at the given point: ( , ) , ( 5, 4)f x y x y. Round the answers to the nearest hundredth. 2. Find the differential of the function. 5sin3tu e x 3. Find the equation of the tangent plane and the normal line to the given surface at the specified point. 32( , ) 4 7f x y x y xy , (1, 1, -3) 4. Find an equation of the tangent plane to the given surface at the specified point. 2218 2 , (3, 2, 1)z x y   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 1. Use the Chain Rule to find sw where s = 3, t = 0. tsztsystxzyxw sin,cos,,222 2. Use the Chain Rule to find pu: zyyxu, trpztrpytrpx 56,56,56 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: )0,1(),sin(),( xyyxf  2. Find the gradient of the function. 2( , , )yf x y z x e z 3. If 229),( yxyxf , use the gradient vector )2,10(f to find the tangent line to the level curve 136),( yxf at the point (10, 2). 4. Find the equation of the tangent plane to the given surface at the specified point 2 2 25 3 8 353, (3, 6, 5)x y z   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 2, 1, 4v (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. 1( , , ) tan , ( 8, 8, 8), 10 7 7yf x y z x vz       i j k 11.7: Maximum & minimum values 1. Find the critical points of the function. 42( , ) 5 76 38 2404yf x y xy x y     2. Find the local maximum, and minimum value and saddle points of the function. 22( , ) 9 6 10f x y x xy y x y      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. 22( , ) 2 2 2 3f x y x y xy x y     subject to 21xy 5. Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In the case of10)1,1(,8)1,1(,7)1,1( yyxyxxfff what can you say aboutf? 6. Find all the saddle points of the function 3sin),(yxyxf  7. Find the absolute maximum value of the function f on the set D. 91083),(222 yxyxyxf,  1||,1|||),(  yxyxD 12.1: Double Integrals over Rectangles 1. Calculate the double integral sin( ) , 0, 0,62x x y dA RR            2. Calculate the iterated integral dxdyyx  6030 3. Calculate the iterated integral ln4 ln5500xye dxdy 4. Calculate the double integral.  2 3 4(3 5 ) , ( , ) | 0 1, 0 4x y x dA R x y x yR      5. Calculate the double integral.  2, ( , )|0 2, 1 124xydA R x y x yxR      6. Calculate the double integral.  , ( , )|0 2,0 1yxye dA R x y x yR     12.2: Double Integrals Over general regions 1. Find the volume of the solid in the first octant bounded by the cylinder 29 yz  and the plane x = 1. 2. Evaluate the iterated integral.  515dxdyxyy3. Evaluate 22Dx y dA where D is the figure bounded by 1, 2, 0,y y x and x y   . 4. Evaluate the double integral dAyD3, where D is the triangular region with vertices (0, 1), (7, 0) and (1, 1). 5. Evaluate the integral by reversing the order of integration.  10442dxdyexy 4. Compute dAyxD224 , where D is the disk ,422 yx by first identifying the integral as the volume of a solid. 12.3: Double Integrals in Polar coordinates 1. Evaluate the integral by changing to polar coordinates: dAeyxD22, where D is the region bounded by the semicircle 24 yx  and the y-axis. 2. Use polar coordinates to find the volume of the solid inside the cylinder 922 yxand the ellipsoid 3622222 zyx. 3. Use polar coordinates to find the volume of the solid under the paraboloid 22yxz  and above the disk 922 yx. 4. Use polar coordinates to find the volume of the solid bounded by the paraboloid 22667 yxz  and the plane z = 1. 5. Evaluate the iterated integral by converting to polar coordinates. dxdyyxy2/3224022)(2 ANSWERS 11.3: Partial Derivatives 1. 4 2 2 415 6 9x x y y 2. 2 2 2 2 2 21,ba b a a b a b    3. 2128 sec 8 tan8y x x 4. 8cos( )x 5. 212 14xyf x y, 212 14yxf x y, 24xxf xy, 14yyfx 11.4: Tangent Planes /Linear Approximations/Differentials 1. 2 1.25 5L x y   2. 555 sin(3 ) 3 cos(3 )ttdu e x dt e x dx 3. 5 10 2x y z   , 1135 10 1x y z   4. 3 4 18x y z   5. 37.54 cm 11.5: The Chain Rule 1. 6 2. 25utpp 11.6: Directional Derivatives & The gradient vector 1. 0, 1 2. 222 , ,2yyyxexe z x e zz 3. 1361810  yx 4. 15 18 40 353x y z   5. a) 35/ 21 b) 4, 11,4    c) 153 6. 5 / (2 198) 11.7: Maximum & minimum values 1. (-6, 6), (-4, 4), (10, -10) 2. Min at (4, 1), (4, 1) 11f  , Max point - none, Saddle point …