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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