Question 1. Show that x^m/n = ⁿ√(x^m) = (ⁿ√x)^m. Specify the rules applied in each step.

Question 2. Let the demand and the supply functions be as follows:
Q_d = 100 -2p
Q_s = -20 + 4p

Find the equilibrium P* and Q* by elimination of variables.

Question 3. Rewrite the market model (problem #2) with variable arranged in the following order Q_d,Q_s, P. Write down the coefficient matrix, the variable vector, and the constant vector. Use Cramer's rule to solve for Q_d*, Q_s*, and P*. Is Q_d*= Q_s*?

Question 4. Given 

Is BC defined? Calculate BC. Is BA defined? Calculate BA. Is BC = BA?

Question 5. Determine if the matrix below is singular or nonsingular.

Question 6. If you found the matrix in problem #6 non-singular, find the inverse matrix, otherwise move on to the next problem. Use fractions rather than decimals.

Question 7. Differentiate the following by using the product rule, make sure to simplify the derivative.

a. (3 - 4x) (1 + x² )( 1 + x )

b. (10x² - 4) x^-1

Question 8. Given y = (10x²+ 3)³, use the chain rule to find dy/dx .

Question 9. Find dy/dx₁ and dy/dx₂ for the following function :

Y = 3x₁⁴ - 12x₁³x² + 3x₁²·x₂²

Question 10. Find the total differential dy of the following function by using rules of differentials:

Y = 2x₁³ + 4x₁²x₂

Question 11. Find the total derivative dz/dy, given

z = 2x² + xy² - x²y where x = y²

Question 12. For the following function, use the implicit- function rule to find dy/dx:

F(x,y) = x³y³ + 3xy

Question 13. Find the stationary values of the following (check whether they are relative maxima or minima or inflection points, assuming the domain to be in the interval (0, ∞).

y = -2x³ + 9x² - 12x + 12

Question 14. Use the chain rule to find dy/dt for the following:

Y = y = 5e^4t

Question 15. Evaluate the following:

(e^{ln u})!

Question 16. Use the Lagrangian-multiplier method to find the stationary value of z:

z = y(x + 10), subject to 2x+y = 40