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 whyHistorically, optimization problems were

among the earliest applications of what we now call differential calculus.

Strategy for Solving Max-Min Problems

Read the problem carefully. Identify the information

you need to solve the problem.

Draw pictures and label the

parts that are im

1. Understand the Problem

2. Develop a Mathematical Model of the Problem

portant 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 t Find the domain of the function. Determine what values

of the variable make sense in the problem.

Find where the derivative is zero or

fails to

he function

4. Identify the Critical Points and Endpoints

exist.

If unsure of the result, support or confirm your

solution with another method.

Translate your mathematical result into the problem setti

5. Solve the Mathematical Model

6. Interpret the Solution ng

and decide whether the result makes sense.

Example 1: Using the StrategyFind 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 ( , sin ) be the coordinates of point and the -coordinate of is ( ).

Thus 2 = length of rectangle and sin = height of rectangle.

( ) 2 sin where 0 / 2. Notice that (0) ( / 2) 0.

x x P x Q x

x x

A x x x x A A

Find the

critical values: '( ) 2sin 2 cos . Let '( ) 0 and use a graphing

calculator to find the solution(s). '( ) 0 at 0.71. The area of the rectangle

is (0.71) 1.12, where the length is 1

A x x x x A x

A x x

A

.72 and its height is 0.65.

Example 3: Fabricating a BoxAn 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 CanYou 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:Terms Formulasx = number of units producedp = price per unitR = total revenue from selling x units R = xpC = total cost of producing x unitsC = the average cost per unitP = total profit from selling P = R - C

The break-even point is the number of units for which R = C.

xf

x

CC

Marginals Marginals

unit) additional one sellingfor profit (extraProfit Marginal

unit) additional one producing ofcost (extraCost Marginal

unit) additional one sellingfor revenue (extra Revenue Marginal

dx

dPdx

dCdx

dR

Examples from EconomicsBig 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 AnalysisBecause 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 6Maximum 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

3 2Suppose that ( ) 9 and ( ) 6 15 , where represents thousands

of units. Is there a production level that maximizes profit? If so, what is it?

r x x c x x x x x

2

2

2

1

2

Set '( ) 9 equal to '( ) 3 12 15.

3 12 15 9

3 12 6 0

Use the quadratic equation to find

2 2 0.586

2 2 3.414

Use a graph to determine that the maximum profit occurs at 3.414.

r x c x x x

x x

x x

x

x

x

Theorem 6Maximizing Profit

Maximum profit (if any) occurs at a production level at which marginal

revenue equals marginal cost.

r’(x) = c’(x)

Theorem 7Minimizing 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 , 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

Solve for x,

Use 2nd derivative test to determine if you’ve found a max or min.

InterpretThe production level to minimize average cost occurs

at x = ____, where x represents thousands of units.

xxxxc 156)( 23

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!