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...

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Discrete Mathematics Examination 2 Question 1 (30 points) (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) b. Find the limit ( ) ( )( )2lim 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 2 (25 points) Given the function: ( ) ( ) ( )2 2, ln lnf x y x x y y x y= + + + . a. Show that ( ) ( )' '1,2 2,1x yf f= . b. 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 3 (20 Points) (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)). b. 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= + + + units 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 4 (25 points) Given : 1 2 3 4 1 2 3 4 1 2 3 4 2 8 0 4 6 2 2 2 10 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. b. Find the values of "k" (if there are any) for which the system has: 1. A unique solution. 2. Infinitely many solutions. 3. No 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
Answered Same DayAug 19, 2021

Answer To: Discrete Mathematics Examination 2 Question XXXXXXXXXXpoints) (there is no connection between the...

Rajeswari answered on Aug 19 2021
121 Votes
63713 Assignment
Since the given point lies on the curve we substitute for x and y to get
Simplify to get
To fin
d y’ first we differentiate giving z =a value
We get
Substitute for y and x the point gives
Simplify to get 0 = 1+
Hence dy/dx = y1 =
b) The given expression can be written as
2ln(-x) +x(k+1)=-2ln (-x) = 10x
When x tends to –infinity first term becomes twice ln of a very large number and hence becomes positively large. But second term being 10x becomes negatively very large.
Since 10x decreases rapidly than increase of 2 lnx we find that the limit final would be –infinity.
c) We find partial derivatives and equate to 0
Equate these two to 0 to get
X=1, y =-4 and x= 2 and y =-3
Critical points are (1,-4) and (2,-3)
No saddle point exists.
We find both are symmetrical
Thus both are equal.
b) g(x) =
Interval given is (-9-2,0) = (-11,0)
Find first two derivatives
When x=-1 I derivative is 0 and second derivative positive.
So minimum at x=-1 and min value is -0.368.
No maxima for this in this interval
c)
This function is defined only for x>0 and hence limit x tends to 0+
= ½ (0) ln (0)
Since x when very...
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