Calculus-Based Optimization Prepared by Lee Revere and John Large Module 6 Calculus-Based Optimization Prepared by Lee Revere and John Large To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-1

Learning Objectives Students will be able to: Find the slope of a curve at any point. Find derivatives for several common types of functions. Find the maximum and minimum points on curves. Use derivatives to maximize total revenue and other functions. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-2

Module Outline M6.1 Introduction M6.2 Slope of a Straight Line M6.3 Slope of a Nonlinear Function M6.4 Some Common Derivatives M6.5 Maximum and Minimum M6.6 Applications To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-3

Introduction There are many business situations in which calculus and derivatives are helpful in finding the best solution to a business problem. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-4

Slope of a Straight Line The equation of a straight line: Y = a + bX Where b is the slope of the line. b = = Given two points: (X , Y ) and (X , Y ) b = Y X Change in Y Change in X 1 1 2 2 Y - Y X - X 1 2 1 2 To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-5

Slope of a Straight Line (continued) 14 12 10 8 6 4 2 -2 -4 Y = -1 + 2X (4,7) Y = 7 – 3 = 4 (2,3) X = 4 – 2 => 2 | | | | 2 4 6 8 To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-6

Slope of a Nonlinear Function If a line is nonlinear, we can find the slope at any point by finding the slope of a tangent line at that point. For example: 2 Y = X - 4X + 6 10 8 6 4 2 -2 0 2 4 6 To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-7

Graph of the Nonlinear Function (continued) 2 Y = X – 4X + 6 20 Slope at point 15 Line through (3,3) and (5,11) Line through (3,3) and (4,6) 10 5 Tangent line at (3,3) -2 2 4 6 Note: As we select points closer to the point where X = 3, we find slopes that are closer to the slope of the tangent line. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-8

Slope of a Nonlinear Function (continued) To find the slope of the tangent line, we must find a value that is VERY close to the X value, thus the change in X should be small. X c aX b Y and Then C a bX D + = - 2 )] ( [ ) 1 To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-9

Common Derivatives The derivative of a function is used to find the slope of the curve at a particular point. Note: The derivative of a constant is 0. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-10

Common Derivatives (continued) The second derivative is the derivative of the first derivative. It tells about the slope of the first derivative. For example: 4 3 Y = 6X + 4X First derivative (Y’) = 6(4)X + 4(3)X Y’ = 24X + 12X Second derivative (Y’’) = 24(3)X + 12(2)X Y’’ = 72X + 24X 4-1 3-1 3 2 3-1 2-1 2 To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-11

Maximum and Minimum A local maximum (or minimum) is the highest (or lowest) point in a neighborhood around that point. A B To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-12

Maximum and Minimum (continued) To find a local optimum, we find the first derivative, set it equal to 0, and solve for X. This is called the critical point. For example: 2 Y’ = X - 8X +12 = 0 Solving for X: (X-2)(X-6) = 0, So the critical points occur at X = 2 and X =6 To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-13

Maximum and Minimum (continued) To find the local maximum and local minimum, we take the second derivative and input the critical points. If the value is negative, we have a local maximum, if the value is positive we have a local minimum. For example: 2 Y’ = X - 8X + 12, thus Y’’ = 2X - 8 Using X = 2: Y’’ = 2(2) – 8 = -4 Using X = 6: Y’’ = 2(6) – 8 = 4 local maximum local minimum To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-14

Maximum and Minimum (continued) A point of inflection occurs when the second derivative is 0. For example: 3 Y = X Y’ = 3X Y’’ = 6X 2 Note: when X = 0 the derivative is zero Note: when X = 0 the derivative is zero To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-15

Maximum and Minimum: Inflection Point To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-16

Maximum and Minimum - Summary A critical point will be: A maximum if the second derivative is negative A minimum if the second derivative is positive A point of inflection if the second derivative is zero To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-17

Applications There are many problems in which derivatives are used in business. Economic Order Quantity: The EOQ model is derived from the derivative of the total cost with respect to Q. Total Revenue: The derivative of the total revenue function yields information on the optimum unit price. To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna M6-18