Reference no: EM131428342

Instructions: The total number of points possible is 100. Show your work for each problem to help us gauge how well you understand the solutions. An answer without showing how you arrived at that answer will not count. Partial credit is available, depending on the work you show. Fundamental mistakes will be much more costly than arithmetic mistakes.

Question 1. Simplify the following expression so that it contains only a single exponent of x.

[2x^-1/2 (3x²)²]³(4x)^-2

Question 2. Differentiate the following function with respect to x:

f(x) = ((9x2 -2)(3x + 1)²)/(2x)

Question 3. Evaluate the derivative of (x+1)/(2x³) at x = 2.

Question 4. Find Y* (equilibrium output) from the following [Hint: Y=(Y^1/2)²]:

Y = C + I₀ +G₀

C = 25 + 6Y^1/2

I₀ = 16

G₀ = 14

Question 5. Find the maximum profit and the quantity (Q) that will maximize the profit function below. Show evidence that profit is indeed maximized:

Π(Q) = 1/3Q³ - 3Q² + 5Q + 3

Question 6. Find the price elasticity of demand of the following:

Q_d = 10/P²

Question 7. Find the derivatives of the following by first taking the natural logs:

a. y = 3x/(x+2)²

b. y = (x² + 3)e^_x2₊₁

Question 8. Find the critical values of the following and use the N^th derivative test to determine if the values determine a maximum, minimum or inflection point.

y = (5 - 2x)⁴ + 8

Question 9. Find the values of x, y and λ that maximize the following utility function subject to the budget constraint: U = (2x² + y²)^1/2; I = 100 = x +2y

Question 10. Given the following matrices:

Find, or if it does not exist state that it does not exist:

a. AD'
b. D'C
c. E'D
d. FB'
e. A^-1C
f. B^-1

Question 11. Consider the following system of equations:

x₁ = 3x₂ + 2x₃ -24

5x₂ = x₃ + 8

x₃ = 6 - x₁

Find x₁ using Cramer's rule.

Question 12. Assume a stock price is a positive function of earnings per share[E], a negative function of bond interest rate [i], and a positive function of inflation [Π] and GDP [Y]. The function is denoted: S = S(E, i, Π, Y). To complicate matters GDP is a negative function of bond interest rate. Write the total differential equation. What can you say about the total derivative of S with respect to the bond interest rate?