Q.1) Express revenue as a function of two quantity demand-price pairs and quantity, assuming that demand price is a linear function, where the quantity demand-pairs are (0,20) and (101,19), and the quantity is 148.
Ans. Revenue (q) =20q-1/101 Q^2
Revenue (148) = 2743.13
Q.2) The function is P1= (9,3) and P2= (3,2)
a. Give the 2 functions of one variable through P1 obtained by holding each variable constant
Ans.
f(9,y) = 81+18y+7y^2
f(x,3) = x^2+6x+63
b. Find the partial derivatives of the original function.
Ans.
f x( x, y ) = 2x+2y
f x( x, y ) = 2x+14y
c. Evaluate the partial derivatives at P1
Ans.
f x( 9, 3 ) = 24
f y( 9, 3 ) = 60
d. Give the equation of the tangent plane through P1
Ans.
f (3, 2) = -6
Q.3) The function is . P1= (8,3) and P2 = (4,9)
a. Give the 2 function of one variable through P1 obtained by holding each variable constant.
Ans.
f(8, y) = 8+7y / 64+y^2
f(x, 3) = x+21 / x^2+9
b. Find the partial derivatives at P1
Ans.
fx = ( x^2 + y^2)-(x+7y)2x / (x^2+y^2)^2
fy=(x^2 =y^2)7 – (x+7y)(2y) / (x^2 + y^2)^2
c. Evaluate the partial derivatives at P1
Ans.
fx (8,3) = -391 / 5329
fy(8,3) =337/ 5329
d. Give the equation of the tangent plane through P1
Ans.
f(x,y)= -391x/5329 +337y/5329 +58/73
e. The approximation at P2 obtained from the tangent line
Ans.
f(4,9) = 5703 /5329
Q.4) The function is P1 = (10,-5) , P2 = (2,5)
a. Give the 2 functions of one variable through P1 obtained by holding each variable constant.
Ans.
f(10,y)=100(10+6^y)
f(x, -5) =x^2 (x+6^-5)
b. Find the partial derivatives of the original function.
Ans.
fx(x,y)=x^2+2x(x+6^y)
fy(x,y)=x^2(6^ylogy)
c. Evaluate the partial derivatives at P1
Ans .
fx(10, -5)=5832051944
fy(10, -5)= (25/1944)log6
d. Give the equation of the tangent plane through P1
Ans.
f(x,y)= 583205x / 1944 +25ylog6 / 1944 – 3888025/1944
e. The approximation at P2 obtained from the tangent plane.
Ans.
f(2,5)=250log6-2721615/1944
Q.5) Given the function f(x,y)=x^2+2xy+10y^2+3x-5y
a. Find the partial derivatives of the original function.
Ans.
fx(x,y)= 2x+2y+3
fy(x,y)= 2x+20y+5
fxx= 2
fxy=2, fyy =20
b. Find the critical point in the region.
Ans. The point is (-25/18, -1/9 )
c. Compute the discriminant at the critical point.
Ans. The discriminant is 36.
d. Determine if the critical point is a maxima, minima or saddle point.
Ans. Maxima.