Lesson 4.4 Modeling and Optimization What you’ll learn about Using derivatives for practical applications in finding the maximum and minimum values in business, economics, and mathematical contexts

Strategy for Solving Max-Min Problems Understand the problem – write down what you are looking for and identify needed information Develop a mathematical model – Draw and label pictures as needed. Write 2 equations. The 1 st will find the extreme value needed. The 2 nd is a constraint and can be used to write the 1 st equation in terms of 1 variable. Use substitution to get a function whose extreme value answers your question. Graph the function. Find the domain and determine what values make sense in the problem. Find f’ of your model & identify the critical points and endpoints. Test to determine if you have maximum or minimum values. Interpret the Solution. Translate your mathematical result into a sentence, confirm that it 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 arc of y = sin x. What is the largest area the rectangle can have, and what dimensions give that area? Draw and label graph. We know A(x) = x y and y = sin x Use substitution to get a function for the area of the rectangle. What dimensions maximize area? Can’t solve algebraically, too messy. Using A(x), find a maximum value on calculator. Using A’(x) = 0, find a maximum value. THEY SHOULD BE THE SAME!!! Interpret your findings: Give maximum length, height, and area.

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 nd 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 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 ____________.

Homework Page 226 Examples 2, 6, 20, 27, 31, 36, 40, 41

Warm Up What is the smallest perimeter possible for a rectangle whose area is 16 in 2, and what are its dimensions?

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)

Example 5: Maximizing Profit Suppose that r(x) = 9x and, where x represents thousands of units. Is there a production level that maximizes profit? If so, what is it? Find r’(x) = c’(x), simplify Find critical points, determine max or min using 2 nd derivative test Interpret Maximum profit occurs when production level is at ________, where x represents thousands of units. Maximum loss occurs when production level is at __________, where x represents thousands of units.

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, 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 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!

Homework Page 226 Exercises 3, 7, 9, 10, 12, 18, 22, 23, 39

Today’s Agenda Present homework problems on the document camera in teams Teamwork: Page 229 #13 work out as a Free Response Question Homework: P