Math 20C Final 9 (No Solutions)

Math 20C Final 9 (No Solutions) - z t = f x t;y t where f...

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Review Problem for 20C, Calculus and Analytic Geometry 1. Use Lagrange multipliers to flnd the maximum and minimum values of the function f ( x;y ) = xy subject to the constraint 2 x + y 2 = 3. 2. Use a triple integral to compute the volume of the tetrahedron whose sides are given by the planes x = 0 ;y = 0 ;z = 0, and 2 x + 2 y + z = 2. 3. Consider the region of D 2 R 3 given by D = f ( x;y;z ) 2 R 3 : x 2 + y 2 1;0 z 1 + x 2 + y 2 g : (a) Sketch the region D . (b) Compute the volume of that region. 4. Let f ( x;y ) = x 3 + y 3 + 12 xy . (a) Find the critical points of f . (b) Use the second derivative test to classify each critical point of f as a local minimum, local maximum or saddle point. 5. Compute R 1 0 R 1 x sin( y 2 ) dydx by flrst changing the order of integration. 6. Transform to polar coordinates and then evaluate the integral I = R 1 ¡ 1 R p 1 ¡ y 2 0 ( x 2 + y 2 ) 3 = 2 dxdy : 7. Find the equation of the plane that contains both the point (-1, 0, 1) and the line x = t;y = ¡ 1 + 2 t;z = 3 t . 8. Consider the function
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Unformatted text preview: z ( t ) = f ( x ( t ) ;y ( t )), where f ( x;y ) = (2 x + y 2 ) 1 = 2 ; x ( t ) = e 3 t ; y ( t ) = e 3 t : Compute d dt z ( t ). 9. Find the linear approximation L ( x;y ) to f ( x;y ) = p x 2 + 3 y 2 at the point (1 : 1 ; 1 : 2). 10. The gradient of a function f ( x;y;z ) at the point (3 ; 4 ; ¡ 5) is (1 ; ¡ 2 ; 2). (a) Find the values of the partial derivatives f x , f y , and f z at the point (3 ; 4 ; ¡ 5). (b) Find an equation for the plane tangent to the graph of f ( x;y;z ) = 0 at the point (3 ; 4 ; ¡ 5). (c) Find the maximum rate of change of f at the point (3 ; 4 ; ¡ 5) and the unit vector in the direction which the maximum rate of change occurs. 11. Find the arc length of the curve r ( t ) = < 2 t 2 + 1 ; 2 t 2 ¡ 1 ;t 3 > over the interval • t • 4....
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