Calculus Answers

The demand for gluten-free pasta is 𝒒𝒅 = 𝟓 + 𝟐𝒆−𝟐𝒑𝒛 + 𝟒√𝒚𝟑 + 𝟑 𝐥𝐧 𝒑−𝟑. Find and interpret the following partial demand elasticities (at 𝒑 = 𝟑, 𝒑𝒛 = 𝟐, 𝒚 = 𝟒):

a. Price elasticity of demand (𝒆𝒑). [2]

b. Cross-price elasticity of demand (𝒆𝒑𝒛 ). [2]

c. Income elasticity of demand (𝒆𝒚). [2]

Check the continuity of the function f: R^2 to R at (0,0)

f(x,y) = { 3x^2y/(x^2+y^2) if (x,y)≠(0,0)

{ 3 if (x,y)= (0,0)

The function f: R^3 to R , defined by

f(x,y,z)= x^3+ e^(y+z) is differentiable everywhere on R^3.

True or false with full explanation

Show that the function f: R^2→R^2 given by

f(x,y) = (xy^3+1, x^2+y^2) is not invertible. Futher check whether it is locally invertible at the point (2,1)

Question 10

A Firm's marginal cost function is Q² + 2Q + 4

Find the total cost function.

a. ^Q3_/3 + Q² + 4 + C

b. ^Q3_/3 + Q² + C

c. ^Q3_/3 + Q² + 4Q + C

d. ^Q3_/3 + Q² + 100

Question  3

Differentiate the function

f (x) = 3x³e^2x.

a. 5x⁴e^3x

b. 3x²e^2x(x − 3)

c. 3e^3x + 9x⁴x⁵

d. 3x²e^2x(2x + 3)

Question  2

Find the derivative of the function Y= x³ + x² /x

a. x2/x + x/x

b. x2+ x /x

c. 2x + 1

d. 3x⁴ + 2x³

Question 15

A firm’s production function is given by

Q = 700Le^−0,02L,

where Q denotes the number of units produced and L the number of labourers. Determine the size of the  workforce that maximises output.

[a.   L = 14

[b.   L = 50

[c.   L = 328

[d.   L = 700

Question 14

Customers of a hardware store are willing to buy

Q = 80 − P ²

boxes of nails at P rand per box. Find the marginal revenue if the price per box is R3.

a.       R−53.00

b.       R71.00

c.       R53.00

d.       R−71.00