2) f(x, y) = 2x2-8y, (4, 8) 7) Find parametric equations for the normal line to the surface z = e8x2 + 4y2 at the point (0, 0, 1).

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1 Exam Name C o m p u t e t h e g r a d i e n t o f t h e f u n c t i o n a t t h e g i v e n p o i n t. ) f(x, y, z) = ln(x + y + z), (,, ) S o l v e t h e p r o b l e m. ) Find the equation for the tangent plane to the surface z = -8x + y at the point (,, -). ) f(x, y) = x - 8y, (, 8) ) Find parametric equations for the normal line to the surface z = e8x + y at the point (,, ). A n s w e r t h e q u e s t i o n. ) Find the direction in which the function is increasing or decreasing most rapidly at the point Po. f(x, y) = xy - yx, Po(-, ) F i n d t h e d e r i v a t i v e o f t h e f u n c t i o n a t t h e g i v e n p o i n t i n t h e d i r e c t i o n o f A. ) f(x, y) = x - y, (-, ), A = i - j F i n d a l l l o c a l e x t r e m e v a l u e s o f t h e g i v e n f u n c t i o n a n d i d e n t i f y e a c h a s a l o c a l m a x i m u m, l o c a l m i n i m u m, o r s a d d l e p o i n t. 8) f(x, y) = xy - x + y ) f(x, y) = x + y - x - y - ) f(x, y, z) = xyz, (,, ), A = - i + j - k

2 ) f(x, y) = - xy ) x exy y F i n d t h e e x t r e m e v a l u e s o f t h e f u n c t i o n s u b j e c t t o t h e g i v e n c o n s t r a i n t. ) f(x, y) = x + y, x + y = ) ex y/8 ) f(x, y) = xy, x + y = E v a l u a t e t h e i n t e g r a l. ) - x x ) f(x, y, z) = x + y + z, x - y - z = 8) y D e t e r m i n e t h e o r d e r o f i n t e g r a t i o n a n d t h e n e v a l u a t e t h e i n t e g r a l. ) tan- x y/

3 ) (s + t) dt ds ) The region bounded by the paraboloid z = x + y, the cylinder x + y =, and the xy-plane F i n d t h e v o l u m e o f t h e i n d i c a t e d r e g i o n. ) The solid cut from the first octant by the surface z = - x - y W r i t e a n e q u i v a l e n t d o u b l e i n t e g r a l w i t h t h e o r d e r o f i n t e g r a t i o n r e v e r s e d. ) - x ) The region that lies under the paraboloid z = x + y and above the triangle enclosed by the lines x =, y =, and y = x ) π/ sin x (x + 8y) ) The region that lies under the plane z = x + y and over the triangle with vertices at (, ), (, ), and (, ) ) ln x x ) tan- x

4 8) y/ ) x/ + ) ln ln y ey E x p r e s s t h e a r e a o f t h e r e g i o n b o u n d e d b y t h e g i v e n l i n e ( s ) a n d / o r c u r v e ( s ) a s a n i t e r a t e d d o u b l e i n t e g r a l. ) The lines x + y =, x + y =, and y = ) x ) The parabola x = y and the line y = x - ) - y ) The parabola y = x and the line y = x + ) y y ) The curve y = ex and the lines x + y = 8 and x = 8 8) The coordinate axes and the line x + y =

5 ) The lines x =, y = x, and y = ) - y (x + y) F i n d t h e a r e a o f t h e r e g i o n s p e c i f i e d b y t h e i n t e g r a l ( s ). ) - x ) - x e-(x + y) ) - y + y ) - - x - - x ( + x + y) C h a n g e t h e C a r t e s i a n i n t e g r a l t o a n e q u i v a l e n t p o l a r i n t e g r a l, a n d t h e n e v a l u a t e. ) - - x ) - - y (x + y) / - - y F i n d t h e a r e a o f t h e r e g i o n s p e c i f i e d i n p o l a r c o o r d i n a t e s. 8) The region enclosed by the curve r = sin θ ) x + x + y

6 ) The region enclosed by the curve r = 8 cos θ ) Integrate f(x, y) = sin(x + y) over the region x + y. ) One petal of the rose curve r = cos θ E v a l u a t e t h e i n t e g r a l b y c h a n g i n g t h e o r d e r o f i n t e g r a t i o n i n a n a p p r o p r i a t e w a y. ) y/ sin x xz dz ) The region enclosed by the curve r = cos θ ) x/ ey z dy dz dx S o l v e t h e p r o b l e m. ) Integrate f(x, y) = ln(x + y) x + y x + y. over the region ) z dz x(y + ) ) Integrate f(x, y) = sin x + y over the region x + y.

7 8) z y + x dy dz dx F i n d t h e v o l u m e o f t h e i n d i c a t e d r e g i o n. ) The region bounded by the coordinate planes and the planes z = x + y, z = E v a l u a t e t h e i n t e g r a l. ) π π π sin(8u + ) du dv dw ) The region in the first octant bounded by the coordinate planes and the surface z = - x - y ) The tetrahedron cut off from the first octant by ) 8 ( - z/8) ( - y/ - z/8) dz the plane x + y + z = ) The region bounded by the paraboloid ) - y z yz dx dz dy z = x + y and the cylinder x + y =

8 ) The region bounded by the cylinder x + y = and the planes z = and x + z = ) Rewrite the integral / ( - z)/ in the order dz. (- y - z)/ dz S o l v e t h e p r o b l e m. ) Write an iterated triple integral in the order dz for the volume of the rectangular solid in the first octant bounded by the planes x =, y =, and z =. ) Write an iterated triple integral in the order dz for the volume of the region enclosed by the paraboloids z = 8 - x - y and z = x + y. 8) Write an iterated triple integral in the order dz for the volume of the region in the first octant enclosed by the cylinder x + y = and the plane z =. ) Write an iterated triple integral in the order dz for the volume of the tetrahedron cut from the first octant by the plane x + y + z =. ) Rewrite the integral 8 ( - x/8) in the order dz. ( - x/8 - y/) dz 8

9 Answer Key Testname: CAL PT ) i + j + k ) i - 8 ) 8 i j ) - ) -.88e+ ) -x + y - z = - ) x =, y =, z = t + 8) f -, = 8, saddle point ) f(, ) = -. local minimum; f(, -) = -, saddle point; f(-, ) = 8, saddle point; f(-, -) = 8, local maximum ) f(, ) =, local maximum ) Maximum: at (, ); minimum: - at (, -) ) Maximum: at, minimum: - at, - ) Maximum: none; minimum: ) tan- - ln, ) e - ) (e - ) ) 8) - ) ) ) 8 ) ) 8π ) ) ) - y π/ (x + 8y) sin- y ln x ey and -, - and -, ; at,-,- ) 8) ) ) ) ) ) ) ) ) ) 8) ) π/ tan y x/ ln x y ln y - x ) ) x y y/(-) ( - y/) ( - y/) y + y x + x ex 8 - x ( - x/) y/ ) 8π π( - ln ) ) ),π 8

10 Answer Key Testname: CAL PT ) π - e - ) π ) π ) ) - - x - - x x + y ( - z/) ( - y/ - z/) dz 8 - x - y dz 8) π ) π ) π ) π ) π(ln ) ) π(sin - cos ) ) π( - cos ) ) ln ( - cos ) ) ln (e - ) ) ln 8) 8 ln ) π cos ) ) - ) ) 8 ) ) 8π ) 8π ) 8) ) ) - / dz - x - - x ( - z/) (- x)/ dz 8( - y/ - z/) dz (- x - y)/ dz