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 im 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 he 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 setti ng 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?

 

x

).

Thus       2

x

   2 sin

x

     2

x

  0 at

x



A

  0. Find the  0 and use a graphing  0.71. The area of the rectangle  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 2 confirm max and min values) nd derivative test to  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 cm 3 Volume formula: Surface Area formula: Use substitution to write an equation.

Solve graphically, confirm analytically. Use 2 nd 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 in 2 , and what are its dimensions?

Business Terms and Formulas

 Business Terms and Formulas: Terms

f

  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

 Marginal Revenue (extra revenue for selling one additional unit)

dx dC

 Marginal Cost (extra cost of producing one additional unit)

dx dP

 Marginal Profit (extra profit for selling

dx

one additional unit)

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

 

x

3  6

x

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

3

x

2 3

x

2   12

x

 15  9  12

x

0  3

x

2  12

x

 15. Use the quadratic equation to find

x

1 2  0.586

x

2 2  3.414

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

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 2 nd 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 2 nd 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!