Discrete Mathematics Examination
2
Question XXXXXXXXXXpoints) (there is no connection between the segments)
a. Given the surface: ( ) ( ), ln 2x yz f x y xe y x−= = + −
A curve is created by intercepting the surface with the plane z a= ("a" is a constant).
Find the value of "a" if the point ( ),e e− − is on this curve, and find ' dyy dx= for this curve
at this point. (you may leave your answer with "e" or use two digits after the decimal)
. Find the limit XXXXXXXXXX2lim ln 1
x
x x k
→−
− + + .
Where the constant "k" represents the last digit of your I.D number.
c. Find extreme points for the function ( ) ( )( ), 2 3 x yf x y x y e += − + and categorize them.
Question XXXXXXXXXXpoints)
Given the function: XXXXXXXXXX, ln lnf x y x x y y x y= + + + .
a. Show that ( ) ( )' '1,2 2,1x yf f= .
. Find global extreme points (if any) for the function ( ) ( ),0xg x f e= on the interval
[ 2,0)k− − , where the constant "k" represents the last digit of your I.D number.
c. Find the limit ( )
0
lim ,0
x
f x
+→
.
d. Find the derivative of the function
( ),0f xy x= at the point x e= .
(you may leave your answer with "e" or use two digits after the decimal)
Shelly Shapiro
3
Question XXXXXXXXXXPoints) (there is no connection between the segments)
a. The profit function from selling x units is : ( )$ 100ln 600 100P x x x= − + .
Find the maximal average profit per unit.
(Decimals values of x can be rounded to the nearest integer (whole number)).
. A marketing manager has found that if he will advertise each month x minutes on Facebook
and y minutes on Instagram, he will sell each month: ( ), 100 3 15f x y x y xy= XXXXXXXXXXunits
when , 15x y .
Choose a point ( ),x y which is positioned on the curve ( ), 2240f x y = , find ' dxx
dy
= for this
curve at this point and explain its business related meaning.
(You may also choose decimals and use two digits after the decimal point).
Question XXXXXXXXXXpoints)
Given :
1 2 3 4
1 2 3 4
1 2 3 4
2 8 0
XXXXXXXXXX
2 6 4
x kx kx kx
x x kx kx k
x kx kx x k
+ + − − =
+ + − = +
+ + − = +
(k is a constant)
a. Is there a value of "k" for which the set is homogenous? If so, find it and if not, explain
why.
. Find the values of "k" (if there are any) for which the system has:
XXXXXXXXXXA unique solution.
XXXXXXXXXXInfinitely many solutions.
XXXXXXXXXXNo solution.
c. Plug 3k = and find the solutions for the system.
d. Choose a value of "k" (other than 3k = ) for which the system has infinitely many solutions
and find the solutions in this case.
e. For the value of "k" you chose in part d: is there a solution in which 1x is equal to the last
digit of your I.D number? Explain.
Shelly Shapiro