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Lesson Plan #27
Date: Wednesday October 31st, 2007
Class: Intuitive Calculus
Topic: Business Applications
Aim: How do we use derivatives to solve business problems?
Objectives:
1) Students will be able to use derivatives to solve business problems.
HW# 27:
1) Given the revenue function R  900 x  0.1x , find the number of units x that produce the maximum revenue.
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2) Find the number of units x that produce the minimum average cost per unit
C , where C 
C
and
x
C  .001x 3  5 x  250
3) Find the price per unit p that produces the maximum profit if the Cost function is
C  100  30x and the demand function
p  90  x
Do Now:
Suppose you were given a cost function C ( X ) and a revenue function R (x ). What would the profit function be?
Suppose a manufacturer can sell
x items a week for a revenue of r ( x)  200 x  0.01x 2 cents, and it costs
c( x)  50 x  20,000 cents to make x items. What is the profit when 7000 items are sold?
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Economists and business people are interested in how changes in variables such as production, price, etc, affect other variables
such as profit, revenue, etc.
Summary of Basic Terms and Formulas:
x is the number of units produced (or sold)
p is the price per unit
R is the total revenue from selling x units.
C is the total cost of producing x units
R  xp
C is the average cost per unit
C
P is the total profit from selling x units
C
x
P  R C
The break-even point is the number of units for which R = C
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Marginals or rates of change with respect to the number of units produced or sold
dR
= Marginal revenue  (extra revenue from selling one additional unit)
dx
dC
 Marginal cost  (extra cost of producing one additional unit)
dx
dP
= Marginal profit  (extra profit from selling one additional unit)
dx
Example #1:
A manufacturer determines that the profit derived from selling
x units of certain item is given by P  0.0002 x 3  10 x .
A) Find the marginal profit for a production level of 50 units?
B) Compare this to the actual gain in profit obtained by increasing the production from 50 to 51 units.
Example #2:
A) A fast-food restaurant has determined that the monthly demand (the number of units x , that consumers are willing to purchase
at a given price p ) for its hamburgers is
p
60,000  x
. Find the increase in revenue per hamburger (marginal revenue) for
20,000
monthly sales of 20,000 hamburgers.
B) Suppose that the cost of producing
x hamburgers is C  5000  0.56x .
Complete the table below
Demand for Hamburgers
20,000
Profit
Marginal Profit
What number of hamburgers sold gives you the maximum profit?
24,400
30,000
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Example #3:
In marketing a certain item, a business has discovered that the demand for the item is represent by the price function
The cost of producing
p
x items is given by C  0.5x  500 . Find the price per unit that yields the maximum profit.
Example #4:
A company estimates that the cost (in dollars) of producing
x units of a certain product is given by
C  800  0.04x  0.0002x . Find the production level that minimizes the average cost per unit.
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50
.
x