AP CALCULUS AB
Chapter 4:
Applications of Derivatives
Section 4.4:
Modeling and Optimization
What you’ll learn about
Examples from Mathematics
 Examples from Business and Industry
 Examples from Economics
 Modeling Discrete Phenomena with
Differentiable Functions

…and why
Historically, optimization problems were
among the earliest applications of what we
now call differential calculus.
Strategy for Solving Max-Min
Problems
1. Understand the Problem Read the problem carefully. Identify the information
you need to solve the problem.
2. Develop a Mathematical Model of the Problem Draw pictures and label the
parts that are important to the problem. Introduce a variable to represent the
quantity to be maximized or minimized. Using that variable, write a function
whose extreme value gives the information sought.
3. Graph the function Find the domain of the function. Determine what values
of the variable make sense in the problem.
4. Identify the Critical Points and Endpoints Find where the derivative is zero or
fails to exist.
5. Solve the Mathematical Model If unsure of the result, support or confirm your
solution with another method.
6. Interpret the Solution Translate your mathematical result into the problem setting
and decide whether the result makes sense.
Example 1: Using the Strategy
Find 2 non-negative numbers whose sum is 20 and whose
product is as large as possible.
What do we want to find?




Identify the variables and write 2 equations. Use
substitution to get a function to maximize.
Simplify, find f ‘, critical points (including endpoints!)
Verify max or min value
Answer question in sentence
You try:
Find 2 numbers whose sum is 20 and the sum of their squares
is as large as possible.
What if we want the sum of the squares as small as possible?
Example 2: Inscribing Rectangles
A rectangle is to be inscribed under one arch of the sine curve. What is the largest
area the rectangle can have, and what dimensions give that area?
Let ( x, sin x) be the coordinates of point P and the x-coordinate of Q is (  x).
Thus   2 x = length of rectangle and sin x = height of rectangle.
A( x)    2 x  sin x where 0  x   / 2. Notice that A(0)  A( / 2)  0. Find the
critical values: A '( x)  2sin x    2 x  cos x. Let A '( x)  0 and use a graphing
calculator to find the solution(s). A '( x)  0 at x  0.71. The area of the rectangle
is A(0.71)  1.12, where the length is 1.72 and its height is 0.65.
Example 3: Fabricating a Box
An open top box is to be made by cutting congruent squares of side length “x” from the
corners of a 20 x 25 inch sheet of tin and bending up the sides.
How large should the squares be to make the box hold as much as possible?
What is the resulting maximum volume?

Draw a diagram, label lengths

V(x) = length • width • height
define length, width, height in terms of x, define domain


Solve graphically, Max of V = Zeroes of V’, confirm analytically
Take derivative, find critical points, find dimensions (use 2nd derivative test to
confirm max and min values)

Find volume, answer question in a sentence, including units.

Example 4: Designing a Can
You have been asked to design a one liter oil can shaped like a right
circular cylinder. What dimensions will use the least material?
What are we looking for?
Given: Volume of can = 1000 cm3
Volume formula:
Surface Area formula:
Use substitution to write an equation.
Solve graphically, confirm analytically. Use 2nd derivative test to confirm min or max.
Answer question in a sentence, include units!
The one liter can that uses the least amount of material has height equal to
____, radius equal to ______ and a surface area of ____________.
You try:
What is the smallest perimeter possible for a
rectangle whose area is 16 in2, and what
are its dimensions?
Business Terms and Formulas
Business Terms and Formulas:
f x
Terms
x = number of units produced
p = price per unit
R = total revenue from selling x units
C = total cost of producing x units
C = the average cost per unit
P = total profit from selling


Formulas
R = xp
C 
C
x
P=R-C
The break-even point is the number of units for which R = C.
Marginals

Marginals
dR
 MarginalRevenue(extrarevenuefor selling one additionalunit)
dx
dC
 MarginalCost (extracost of producingone additionalunit)
dx
dP
 MarginalProfit(extraprofitfor selling one additionalunit)
dx
Examples from Economics
Big Ideas
 r(x) = the revenue from selling x items
 c(x) = the cost of producing x items
 p(x) = r(x) – c(x) = the profit from selling x items
Marginal Analysis
Because differentiable functions are locally linear, we use the marginals to approximate
the extra revenue, cost, or profit resulting from selling or producing one more item. We
find the marginal analysis by taking the derivative of each function.
Theorem 6
Maximum profit (if any) occurs at a production level at which marginal revenue equals
marginal cost.
 p’(x) = r’(x) – c’(x) is used to find the production level at which maximum profit
occurs (Theorem 6)
Maximum Profit
Maximim profit (if any) occurs at a production level at which marginal
revenue equals marginal cost.
Example Maximizing Profit
Suppose that r ( x)  9 x and c( x)  x  6 x  15 x, where x represents thousands
3
2
of units. Is there a production level that maximizes profit? If so, what is it?
Set r '( x)  9 equal to c '( x)  3 x  12 x  15.
3 x  12 x  15  9
2
2
3 x  12 x  6  0
Use the quadratic equation to find
2
x  2  2  0.586
1
x  2  2  3.414
2
Use a graph to determine that the maximum profit occurs at x  3.414.
Theorem 6
Maximizing Profit
Maximum profit (if any) occurs at a
production level at which marginal
revenue equals marginal cost.
r’(x) = c’(x)
Theorem 7
Minimizing Cost
The production level (if any) at which
average cost is smallest is a level at
which the average cost equals the
marginal cost.
c’(x) = c(x) / x
Example 6: Minimizing Average Cost
Suppose c( x)  x  6x  15x , where x represents thousands of units.
Is there a production level that minimizes average cost? If so, what is
it?
We want c’(x) = c(x) / x
3
2
Solve for x,
Use 2nd derivative test to determine if you’ve found a max or min.
Interpret
The production level to minimize average cost occurs at x = ____, where x
represents thousands of units.
Summary - How can we solve an
optimization problem?
Identify what we want to find and the information we are
given to find it.
 Draw a picture, write equations, use substitution to get a
function in terms of the variable needed.
 Solve graphically, confirm analytically
 Find max / min points on the graph, don’t forget to consider
endpoints. Use 2nd derivative test to confirm max or min.
 Use the values you’ve found to answer the original question
in a sentence. Make sure your answer makes sense!
