MATH-1326-THQ 8, 2016-03-07 12:35 (Printed First Name) (Printed Last Name) XXX0000 Net ID (First Letter of Last Name) 001 - 503 Section Take Home Quiz 8 Penalty Instructions 1. Fill in the requested information on the line above. 2. This handout is due at the beginning of lecture on Tuesday (03-22-2016). One point penalty per minute late. Submit right away, don't wait for the end of class! THQ with missing name will receive 10 points penalty. 3. This handout must be printed out and. You may print it single sided or double sided. Failing to print costs 10 points. 4. This handout must be stapled. Failing to staple costs 10 points. 5. Your work must be hand written on this handout. 6. You must show all work. You may receive zero or reduced points for insucient work. 7. Your work must be neatly organized and written. You may receive zero or reduced points for sloppy work. 8. Only a subset of these questions will be graded. You will not be told which questions will be graded in advance. Due Date: Tuesday, 03-22-2016 Page 2 of 9 MATH-1326-THQ 8, 2016-03-07 12:35 Problem 1 Find all critical points for the following functions. Classify each critical point as a relative minimum, relative maximum or saddle points. (a) f (x, y) = x + 9 4 y + 25 x y Page 3 of 9 (b) f (x, y) = ex 2 2x+y 2 MATH-1326-THQ 8, 2016-03-07 12:35 Page 4 of 9 Problem 2 (a) Find relative extrema of the function f (x, y) = xy subjected to the constraint g(x,y) = x + y - 200 = 0. (b) Find relative extrema of the function f (x, y) = 3x2 + 4y 2 subjected to the constraint x+2y = 9. MATH-1326-THQ 8, 2016-03-07 12:35 Page 5 of 9 Problem 3 Find relative maximum value of the function f (x, y) = x2 y subjected to the constraint g(x,y) = x + y - 10 = 0. MATH-1326-THQ 8, 2016-03-07 12:35 Page 6 of 9 MATH-1326-THQ 8, 2016-03-07 12:35 Problem 4 A manufacturing rm estimates that its total production of automobile batteries in thousands of units is given by, 1 1 f (x, y) = 3x 2 y 2 where x is the number of units of labor and y is the number of units of capital utilized. Labor costs are 70 per unit, and capital costs are 60 per unit. How many units each of labor and capital will maximize production, if the rm can spend 42, 000 for these costs? This problem can be translated in the following optimization problem. Find relative extrema of the function 1 1 f (x, y) = 3x 2 y 2 subjected to the constraint 70x + 60y = 42000. Page 7 of 9 MATH-1326-THQ 8, 2016-03-07 12:35 Problem 5 A rectangular box with open top is to be built from 1200 cm2 of material. Find the dimension of such a box that will enclose the maximum volume. This problem can be translated in the following optimization problem. Maximize, f (x, y) = xyz subjected to the constraint g(x, y, z) = 2xz + 2yz + xy - 1200 = 0. Page 8 of 9 Problem 6 Evaluate following integrals: 1 9x2 + 7y 2 dx (a) 0 1 2x2 + 6y 2 dy (b) 0 MATH-1326-THQ 8, 2016-03-07 12:35 Page 9 of 9 Problem 7 Evaluate following integrals: 1 1 12x2 y + 6y 2 dx dy (a) 1 0 2 2 6x2 + 3y 2 dy dx (b) 0 0 MATH-1326-THQ 8, 2016-03-07 12:35 \f\f\f\f\f\f\f\fThe denominator provides the restrictions that 0