Education.com
Try
Brainzy
Try
Plus

Profit, Revenue and Cost Problems for AP Calculus

based on 1 rating
By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Applications of Derivatives Practice Problems for AP Calculus

Summary of Formulas

  1. P = RC: Profit = Revenue – Cost
  2. R =xp: Revenue = (Units Sold)(Price Per Unit)
  3. = Marginal Revenue ≈ Revenue from selling one more unit
  4. = Marginal Profit ≈ Profit from selling one more unit
  5. = Marginal Cost ≈ Cost of producing one more unit

Example 1

Given the cost function C(x) = 100 + 8x + 0.1x2,

  1. find the marginal cost when x = 50; and
  2. find the marginal profit at x = 50, if the price per unit is $20.

Solution:

  1. Marginal cost is C(x). Enter d(100 + 8x + 0.1x2, x)|x = 50 and obtain $18.
  2. Marginal profit is P '(x)
  3. P = RC

    P = 20x – (100 + 8x + 0.1x2). Enter d(20x – (100 + 8x + 0.1x^ 2, x)|x = 50 and obtain 2.

Example 2

Given the cost function C(x) = 500 + 3x + 0.01 x2 and the demand function (the price function) p(x) = 10, find the number of units produced in order to have maximum profit.

Solution:

Step 1:  Write an equation.

    Profit = Revenue – Cost
    P = RC
    Revenue = (Units Sold)(Price Per Unit)
    R =xp(x) = x(10) = 10x
    P = 10x – (500 + 3x + 0.01x2)

Step 2:  :Differentiate.

    Enter d(10x – (500 + 3x + 0.01x^2, x) and obtain 7 – 0.02x.

Step 3:  Find critical numbers.

    Set 7 – 0.02x = 0 x = 350.
    Critical number is x = 350.

Step 4:  Apply Second Derivative Test.

    Enter d(10x – (500 + 3x + 0.01x^2), x, 2)|x = 350 and obtain –0.02.
    Since x = 350 is the only relative maximum, it is the absolute maximum.

Step 5:  Write a Solution.

    Thus, producing 350 units will lead to maximum profit.

Practice problems for these concepts can be found at: Applications of Derivatives Practice Problems for AP Calculus

Add your own comment