# 1 2 f f x y ye x 0 4 2 3 f x y e x cos y 0 0 4 3 4 a

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1 . , 2 . f f x , y ye x 0, 4 , 2 3 f x , y e x cos y , 0, 0 , 4 3-4 (a) Find the gradient of f . (b) Evaluate the gradient at the point P . (c) Find the rate of change of at f P in the direction of the 3 . 4 . , , 5-7 Find the directional derivative of the function at the given point in the direction of the vector v . 5 . , 6 . 7 . vector . u f x , y sin 2 x 3 y , P 6, 4 , u 1 2 ( s 3 i j ) f x , y , z x 2 y z xy z 3 P 2, 1, 1 u 0, 4 5 , 3 5 f x , y e x sin y , 0, 3 v 6, 8 t p , q p 4 p 2 q 3 , 2, 1 , v i 3 j f x , y , z xe y ye z z e x , 0, 0, 0 , v 5, 1, 2 14.7 1-4 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 1 . 2 . 3 . 4 . f x , y xy 2 x 2 y x 2 y 2 f x , y xe 2 x 2 2 y 2 f x , y xy 1 x y f x , y sin x sin y , x , y 5-7 Find the absolute maximum and minimum values of f on the set . 5 . , is the closed triangular region with vertices 6 . , 7 . , D f x , y x y xy D 0, 0 , 0, 2 , and 4, 0 f x , y 4 x 6 y x 2 y 2 , D x , y 0 x 4, 0 y 5 f x , y xy 2 D x , y x 0, y 0, x 2 y 2 3 8 . to the plane Find the shortest distance from the point x y z 1. 9 . that is closest to Find the point on the plane the point 0, 1, 1 . 2, 0, 3 x 2 y 3 z 6 45. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r .
14.8 1-5 Use Lagrange multipliers to find the maximum and mini- mum values of the function subject to the given constraint. 2 . ; 3 . ; 4 . ; 5 . 6-7 Find the extreme values of f subject to both constraints. 6 . 1 . f x , y x 2 y 2 ; xy 1 f x , y y 2 x 2 1 4 x 2 y 2 1 f x , y , z 2 x 2 y z x 2 y 2 z 2 9 f x , y , z xy z x 2 2 y 2 3 z 2 6 f x , y , z x 2 y 2 z 2 ; x 2 y 2 z 2 1 f x , y , z x 2 ; y x y z 1, y 2 z 2 4 7 . 8 Find the extreme values of f on the region described by the inequality. f x , y , z x 2 y 2 z 2 ; x y 1, y 2 z 2 1 f x , y 2 x 2 3 y 2 4 x 5, x 2 y 2 16 15.1 1 . If R 0, 4 1, 2 , use a Riemann sum with , . Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles. m 2 n 3 to estimate the value of xx R 1 xy 2 dA 2 . (a) Estimate the volume of the solid that lies below the surface z 1 x 2 3 y and above the rectangle R 1, 2 0, 3 . Use a Riemann sum with m n 2 and choose the sample points to be lower left corners. (b) Use the Midpoint Rule to estimate the volume in part (a). 3 6 Calculate the iterated integral. 5 . 6 . 4 . 1 0 3. y 4 y 2 6 x 2 y 2 x dy dx y 4 1 y 2 1 x y y x dy dx y 1 0 y 1 0 v u v 2 4 du d v y 1 0 y 1 0 s s t ds dt 7-9 Calculate the double integral. 7 . yy R y xy 2 dA , R x , y 0 x 2, 1 y 2 8 . 9 . yy R x sin x y dA , R 0, 6 0, 3 yy R ye xy dA , R 0, 2 0, 3 10. Find the average value of over the given rectangle. , f f x , y e y s x e y R 0, 4 0, 1 15.2 1-2 Evaluate the iterated integral. 2 . 3-5 Evaluate the double integral. 3 . 4 . 5 . 0 y 1 . y 2 y 2 y xy dx dy y 1 0 y s 2 0 cos s 3 dt ds yy D yy D yy D y 2 dA , D x , y 1 y 1, y 2 x y x dA , D x , y 0 x , 0 y sin x x 3 dA , D x , y 1 x e , 0 y ln x 6-8 Evaluate the double integral. 6 . is bounded by 7 . 2 x y dA , is bounded by the circle with center the origin and radius 2 8 . is the triangular region with vertices , , and yy D x cos y dA , D y 0, y x 2 , x 1 yy D D yy D 2 xy dA , D 0, 0 1, 2 0, 3 9-12 Find the volume of the given solid.
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