Assume that a competitive firm has the total cost function:

TC=1q

3

40q

2

+880q+2000

TC=1q3-40q2+880q+2000

Suppose the price of the firm's output (sold in integer units) is $550 per unit.Use calculus and formulas to find a solution (don't just build a table in a spreadsheet as in the previous lesson).Hint 1:The first derivative of the total profit function, which is cumulative, is the marginal profit function, which is incremental. The lecture and formula summary explain how to compute the derivative.Set the marginal profit equal to zero to define an equation for the optimal quantity q.Rearrange the equation to the quadratic form aq2+ bq + c = 0, where a, b, and c represent numbers.Use the quadratic formula to solve for q:

q=bb

2

4ac

2a

q=-bb2-4ac2a

For non-integer quantity, round up and down to find the integer quantity with the optimal profit.Hint 2:When computing the total profit for each candidate quantity, use the total profit function you define (rather than summing the marginal profits using the marginal profit function).

How many integer units should the firm produce to maximize profit?Please specify your answer as an integer.

What is the total profit at the optimal integer output level?Please specify your answer as an integer.

Suppose a competitive firm has as its total cost function:

TC=23+3q

2

TC=23+3q2

Suppose the firm's output can be sold (in integer units) at $72 per unit.Use calculus and formulas to find a solution (don't just build a table in a spreadsheet as in the previous lesson).Hint 1:The first derivative of the total profit function, which is cumulative, is the marginal profit function, which is incremental. The lecture and formula summary explain how to compute the derivative.Set the marginal profit equal to zero to define an equation for the optimal quantity q.Hint 2:When computing the total profit for a candidate quantity, use the total profit function you define (rather than summing the marginal profits using the marginal profit function).

How many integer units should the firm produce to maximize profit?Please specify your answer as an integer.In the case of equal profit from rounding up and down for a non-integer initial solution quantity, proceed with the higher quantity.

What is the total profit at the optimal integer output level?Please specify your answer as an integer.

Assume that a monopolist faces a demand curve for its product given by:

p=802q

p=80-2q

Further assume that the firm's cost function is:

TC=560+13q

TC=560+13q

Use calculus and formulas to find a solution (don't just build a table in a spreadsheet as in the previous lesson).Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should also round to the nearest hundredth. Use these rounded values to compute optimal profit. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items.Hint 1:Define a formula for Total Revenue using the demand curve equation.Hint 2:The first derivative of the total profit function, which is cumulative, is the marginal profit function, which is incremental. The lecture and formula summary explain how to compute the derivative.Set the marginal profit equal to zero to define an equation for the optimal quantity q.Hint 3:When computing the total profit for a candidate quantity, use the total profit function you define (rather than summing the marginal profits using the marginal profit function).

How much output should the firm produce?Please round your answer to the nearest hundredth.

What price should the monopolist choose to maximize profits?Follow the rounding guidance in the exercise statement for the optimal quantity when you compute the optimal price. Please round your optimal price answer to the nearest hundredth.

What is the profit for the firm at the optimal quantity and price?Follow the rounding guidance in the exercise statement for quantity and price when you compute the optimal profit. Please round your optimal profit answer to the nearest integer.