Q1

For the given function f(r, y), find fr(x, y) and fy (x, y) f(x, y) = 3rely Ofx(x, y) = 3ery' (1 + xy?), fy(x, y) = 6x2yery O fr(x, y) = 3ev (1 + xy?), fy(x, y) = 6x2 yery O fr(x, y) = 3ery' (1 + xy? ), fy(x, y) = 6x23ery Ofx(x, y) = 3y? ery', fu(x, y) = 6x2 yery Of,(x, y) = 3y?ery', fy (x, y) = 6x2y?ery Of:(x, y) = 3es' , fy (x, y) = 3xezzyEvaluate the partial derivative fr(x, y) and fy(x, y) at the point (0, -1). f(I, y) y 2x + y O fx(0, -1) = 4, fy(0, -1) = 0 O fr(0, -1) = 2, fy(0, -1) = -2 O fx(0, -1) = 0, fy(0, -1) = -3 O fx(0, -1) = -3, fy(0, -1) =4 O fr(0, -1) = 2, fy(0, -1) =0 O fx(0, -1) = -2, fy(0, -1) =-2For the given function f(x, y), find fry (x, y). f(x, y) = =ty -cy O fry (I, y) = (x2 + 2)3/2 O fry (I, y) = (a2 + y2)2 O fry ( I, y) = ry (a2 + 72)3/2 O fry (I, y) = 12 y2 (x2 + y2)2 O fry (I, y) = -xy (a2 + y2)3 O fry(I, y) : cy (a2 + y2)3Find the critical point(s) of the given function f(I, y). 16 6 f(I, y) = + 3y? y O (0, 0) and (2, -1) O There are no critical points of f(I, y) O (8, -6) O (2, 1) O (0, 0) and (8, -6) O (2, -1)Two of the critical points of the following function are (1, 1) and (1, -1) Classify each as a relative maximum, relative minimum, or saddle point. f(x, y) = 2x3 +y3 + 3x2 -3y - 12x -4 O (1, 1) and (1, -1) are both relative minimums O (1, 1) and (1, -1) are both saddle points O (1, 1) is a saddle point and (1, -1) is a relative maximum O (1, 1) is a relative maximum and (1, -1) is a saddle point O (1, 1) is a relative minimum and (1, -1) is a relative maximum O (1, 1) is a relative minimum and (1, -1) is a saddle pointA company can produce green and red notebooks, each costing $2 each to produce. Research has estimated that is the green notebooks are sold for 2 dollars and red notebooks for y dollars, then they can sell 40 - 50x + 40y of the green notebooks, and 20 + 60x - 70y of the red notebooks. What should the company charge for the green notebooks to maximize profit? O $2.00 O $2.10 O $2.50 O $2.25 O $2.70 O $3.00Given f(x, y) = 2x2 + 4y' -3xy -2x - 23y + 3, find the minimum value of f(T, y) subject to the constraint x + y = 15. O -10 O -22 O -27 O -35 O -18 0 0A company can produce 2 types of lamps, type A and type B. It is estimated that when they produce x units of type A and y units of type B per month, the monthly profit will be P(x, y) - 0.3x2 - 0.5xy - 0.4y' + 85x + 125y - 2500 If the company can produce 300 lamps per month, how many lamps of Type B should be produced to maximize profits? O 200 O 150 O 100 O 75 O 175 O 125Evaluate the definite integral. In 2 2xey dxdy -1 O 10 0 4 0 2 0 -1 0 5 0 -7Evaluate the definite integral. x2 ydydr 0 8 O 36 O 32 O 24 0 4 O 16Which of the following integrals could be used to calculate the volume of the solid below the surface f(x, y) = 3ry bounded by the region R which is a triangle with vertices at (0, 0), (2, 0), and (2, 6). o fo fo 3ry dydr o fo fo 3ry dady o fo fo 3ry dady o So ful2 3ry- dyda o fa So 3ry dyda o fa S 3ry dydaEvaluate the definite integral. (HINT: Change the order of integration). 1 dxdy y 0 0 O 4 In 2 O In 2 O 1In 2 O In2 - 1 O 2 In 2