### 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.