MAT 211 Quiz 1 solutions
1.(6pts) Use the following contour plot to answer the questions below.
See figure on your quiz.
(a) f (x,1) = 4 is true for what x value(s)?
x=2 (approx) and x=3
(b)What is the sign of f x(2,4) ? Explain why.
it is positive because moving from the point (2,4) in the positive x direction we get from level 2 to level 4.
(c)What is the highest possible value for f (4,y)? At what (x,y) point it is happening?
It is 6 and it is happening at the point (4,3)
- (6pts) Using x skilled workers and y unskilled workers, a manufacturer can produce Q(x,y) = 10x2y units per day.
(a)Find Q(20, 40) and interpret your answer with a complete English sentence.
Q(20, 40)= 10(20)2(40)= 160,000 . Using 20 skilled workers and 40 unskilled workers, the manufacturer can produce 160,000units per day.
(b)Find Qx(20,40) and interpret your answer with a complete English sentence.
Qx(x,y)= 20xy so Qx(20,40) = 20(20)(40) = 16,000. If the number of skilled workers increases from 20 to 21 and if the number of unskilled workers is fixed at 40 then the manufacturer can produce 16,000 MORE units per day.
(c)Find Qy(20,40) and interpret your answer with a complete English sentence.
Qy(x,y)= 10x 2 so Qx(20,40) = 10(20)2 = 4,000. If the number of skilled workers is fixed at 20 and if the number of unskilled workers increases from 40 to 41 then the manufacturer can produce 4,000 MORE units per day.
3.(8 pts)Use the function f (x,y) = x3 - y3 + 6xy to answer the following questions.
(a)find the first partial derivatives
f x(x,y) = 3x 2 + 6y
f y(x,y) = -3y 2 + 6x
(b)given that the two stationary points of this function are (0,0) and
(2, -2) classify them. Show all your work.
f xx(x,y) = 6x
f yy(x,y) = -6y
f xy(x,y) =6= f yx(x,y)
D = f xx(x,y) · f yy(x,y) - [f xy(x,y)]2 = 6x · (-6y ) - 36
point / D / f xx(x,y) / Classification(0,0) / 0(0)- 36 < 0 / Saddle point
(2,-2) / 12(12)- 36 > 0 / 6(2) > 0 / Local minimum