1. Take the derivative of the following functions with respect to x:

f(x)=x^4-4x^3

A. Y=Ax^3+6x^2-Bx, Where A and B are unknown constants.

B. f(x)=x^3(x-2)^2

C. Y=3x^4-x+4-x^-1+x^2/3-x^-5+x^7/5

2. Graph the following functions in a (Q, p) space, i.e., Q is in horizontal axis, and p in vertical axis.

a. Q= 70+ 4p. What is the intercept on p and Q axis, respectively? What is the slope?

b. p = 5 - 2Q. What is the intercept on p axis and Q axis, respectively? What is the slope?

1. The intercept of a line is 5 on x axis (horizontal), and the intercept is 20 on y axis (vertical). Write down the function of this line (express y as a function of x).

2. The y-intercept of a line is 4, and the slope of the line is -0.6. Write down the function of this line.

3. Dynamic Corp. sells its output at the market price of $13/unit. Each plant has the costs shown below. How much output should each plant produce? Units of Output Total Cost ($)

Units Total Cost ($)

of Output

0 6 1 9 2 15 3 24 4 36 5 51 6 - 69 7 90

5. The demand function of a company is Q = 20 - 0.4p.

a. Please derive its inverse demand function, i.e., express p as a function of Q. (Q stands for quantity demanded, and p is the market price).

b. What is the marginal revenue function?

c. If the marginal cost of the company is $10. What is the optimal quantity that maximizes its profit? What is the maximized profit?

d. Graph the marginal cost and marginal revenue function in (Q, p) space. Indicate their intercepts on each axis. Indicate the optimal output and price in the graph.

e. Suppose the marginal cost decreases to $8. What is the optimal quantity? Does it increase, decrease, or stay the same? In the same diagram from (5), graph the change of marginal cost function, optimal quantity, and optimal price (if there is any change at all).

f. Suppose that you can sell as much of a product as you like at $92 per unit. Your marginal cost (MC) for producing the Qth unit is given by: MC=10Q. If fixed costs are $350, what is the optimal output level?

g. Suppose a firm has as its total cost function: TC = 24 + 2Q2, and its output can be sold at $44 per unit. Using calculus to find the firm's profit-maximizing output. What is the maximized profit?