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final_99 - Math 100 Introduction to Multivariable Calculus...

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Math 100 - Introduction to Multivariable Calculus FINAL EXAMINATION Fall Semester, 1999 Time Allowed: 2.5 Hours. Total Marks: 100 Student Name: Student Number: 1. (10 marks) Locate all relative maxima, relative minima and saddle points of the function f ( x, y ) = x 3 + y 3 - 3 x - 3 y. 2. (10 marks) Evaluate the double integral ZZ D 1 (1 + x 2 + y 2 ) 2 dxdy, where D = n x 2 + y 2 R 2 fl fl fl x 0 , y 0 o . 3. (15 marks) Evaluate the line integral I C ( e x 2 + y 2 ) dx + ( e y 2 + x 2 ) dy where C is the boundary of the region between y = 2 x 2 and y = 2 x and is oriented counterclockwise. 4. (15 marks) Evaluate the line integral Z C (sin x + sin y ) dx + (1 + x cos y ) dy, where C is the line segment from (0 , 0) to ( π, π ). 1
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Math 100 - Introduction to Multivariable Calculus 2 5. (15 marks) Find the flux of ~ F across the surface of conical solid bounded by z = q x 2 + y 2 and z = 1 , where ~ F ( x, y, z ) = x 2 ~ i + y 2 ~ j + z 2 ~ k. 6. (15 marks) Find the volume lying inside both the sphere x 2 + y 2 + z 2 = a 2 and the cylinder x 2 + y 2 = ax , where a is a positive constant. 7. (20 marks) Use the Divergence Theorem to evaluate the surface integral ZZ σ ~ F ( x, y, z ) · ~n dS, where ~ F ( x, y, z ) = 2 x 2 ~ i + ( y + z - 4 xy ) ~ j + ~ k and σ is the surface of the upper-hemisphere z = q 1 - x 2 - y 2 , z > 0 without the bottom disk (i. e., not including the disk x 2 + y 2
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