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Find the absolute maximum and absolute minimum values of f(x,y)=(x-3)² + y² D: {(x,y):0≤x≤ 4, x² ≤ y ≤ 4x}

Find the absolute maximum and absolute minimum values of f(x,y)=(x-3)² + y² D: {(x,y):0≤x≤ 4, x² ≤ y ≤ 4x}

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8. Find the absolute maximum and absolute minimum values of f(x, y) = (x−3)² + y² on the set D: {(x,y): 0≤x≤ 4, x² ≤ y ≤ 4x} 9. a. An open-topped rectangular container is to have a volume of 32 cubic meters. Find the dimensions of the container having the smallest surface area. b. The temperature T'at a point (x, y, z) on the sphere x² + y² +²=1 is given by T(x, y, z)=400xyz². What are the maximum and minimum temperatures? 10. Change the order of integration: 12. 2 ny · [[ f(x, y)dxdy In4 4 b. fff(x,y)dydx 11. Set up a double integral to find the volume of the solid S in the first octant that is bounded above by the surface ==9-x² - y², below by the xy-plane, and on the sides by the planes x-0 and y=x. a. Evaluate , (x²+2=)dxdydz where I is the region bounded by the planes z=0 and y+z=4 and the cylinder y=x² b. Set up a triple integral in cylindrical coordinates that gives the volume of the solid bounded above by the hemisphere ==√√2-²-² and below by the paraboloid == x² + y² c. Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone = √3x² +3y² and inside the hemisphere ==√√4x² - y² y-x 13. Evaluate sin y+x and .x+y=2. (Use the Jacobian with u=y-x and v= y + x) dxdy where is the region in the first quadrant bounded by the line x+y=1 14. Calculate h(r) dr where h(x, y) = (6xy - y²)i + (4y+3x²-3xy³)j and C is the curve consisting of the line segment from (0, 0) to (2, 4) and the parabola y=x² from (2, 4) to (3,9).
Transcribed Image Text:8. Find the absolute maximum and absolute minimum values of f(x, y) = (x−3)² + y² on the set D: {(x,y): 0≤x≤ 4, x² ≤ y ≤ 4x} 9. a. An open-topped rectangular container is to have a volume of 32 cubic meters. Find the dimensions of the container having the smallest surface area. b. The temperature T'at a point (x, y, z) on the sphere x² + y² +²=1 is given by T(x, y, z)=400xyz². What are the maximum and minimum temperatures? 10. Change the order of integration: 12. 2 ny · [[ f(x, y)dxdy In4 4 b. fff(x,y)dydx 11. Set up a double integral to find the volume of the solid S in the first octant that is bounded above by the surface ==9-x² - y², below by the xy-plane, and on the sides by the planes x-0 and y=x. a. Evaluate , (x²+2=)dxdydz where I is the region bounded by the planes z=0 and y+z=4 and the cylinder y=x² b. Set up a triple integral in cylindrical coordinates that gives the volume of the solid bounded above by the hemisphere ==√√2-²-² and below by the paraboloid == x² + y² c. Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone = √3x² +3y² and inside the hemisphere ==√√4x² - y² y-x 13. Evaluate sin y+x and .x+y=2. (Use the Jacobian with u=y-x and v= y + x) dxdy where is the region in the first quadrant bounded by the line x+y=1 14. Calculate h(r) dr where h(x, y) = (6xy - y²)i + (4y+3x²-3xy³)j and C is the curve consisting of the line segment from (0, 0) to (2, 4) and the parabola y=x² from (2, 4) to (3,9).
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