Math 3
Introduction to Calculus
Bjoern Muetzel
Department of Mathematics
Dartmouth College
Lecture 20 - 2/17/2016
Optimization II
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Example:
You have a small tattoo shop. The cost
of inking x tattoos in a month is
C(x) = x2/4 +40x + 400,
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0 < x < 80.
The price of a tattoo is about 70 USD.
Find the number of tattoos you should
ink to maximize your profit.
Summary: Finding extremal values has many practical applications. We
can maximize area or volume, minimize distance and time, minimize cost
and maximize profit.
Lecture 20 - 2/17/2016
Application to economics
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1.) Cost function C(x):
The function C(x) that represents the total cost C for producing x units of a product
is called a cost function.
C'(x), its derivative is the rate of change of the cost. It is called the marginal cost.
2.) Revenue function R(x):
The function R(x) that represents the total revenue R for x units of a product is
called the revenue function.
R'(x), its derivative is called the marginal revenue.
3.) Profit function P(x):
The function
P(x) = R(x) – C(x)
is called the profit function. Its derivative is called the marginal profit.
4.) A usual goal of an enterprise is to maximize the profit P(x).
Lecture 20 - 2/17/2016
Optimization - summary
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1.) Read the problem carefully and draw a diagram.
2.) Assign symbols to the various quantities of the problem and write down
their relations and additional constraints.
3.) Express the given quantity to be maximized/minimized in terms of other
quantities.
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4.) If there is more than one variable, eliminate all but one variable.
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5.) Find the absolute maximum/minimum using the various tests studied.
Lecture 20 - 2/17/2016
To do
1. I will group you in groups of three. Introduce yourself, then:
2. Do exercise 19 together and write down a proper solution in
class. Do as many problems as you can, but take your time.
3. Time to prepare for the Midterm exam II. You can prepare
with the practice exams on the course page. Take a close
look at the solutions of the longer problems.
If you have questions that arise, please mail them to me from
within Canvas or to my Dartmouth email.
Lecture 20 - 2/17/2016