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Copyright © by Houghton Mifflin Company, All rights reserved. Chapter # Key Concepts Approximating Change Using TangentsApproximating Change Using TangentsApproximating Change Using TangentsApproximating Change Using Tangents ExtremaExtremaExtrema Optimization Without EquationsOptimization Without EquationsOptimization Without EquationsOptimization Without Equations Inflection PointsInflection PointsInflection PointsInflection Points Related RatesRelated RatesRelated RatesRelated Rates

Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Change Using Tangents The change in the tangent line from x to x+h is (slope of tangent line at x) hThe change in the tangent line from x to x+h is (slope of tangent line at x) h The change in a function from x to x+h can be approximated by the change in the tangent line from x to x+h when h is a small numberThe change in a function from x to x+h can be approximated by the change in the tangent line from x to x+h when h is a small number That is, f(x+h) - f(x)  f '(x) hThat is, f(x+h) - f(x)  f '(x) h The result of the change in input from x to x+h is f(x+h)  f(x) + f '(x) hThe result of the change in input from x to x+h is f(x+h)  f(x) + f '(x) h

Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Change Using Tangents Marginal Cost: the rate of change of cost with respect to the number of units produced. That is, C'(x).Marginal Cost: the rate of change of cost with respect to the number of units produced. That is, C'(x). Marginal Revenue: the rate of change of revenue with respect to the number of units sold. That is, R'(x).Marginal Revenue: the rate of change of revenue with respect to the number of units sold. That is, R'(x). Marginal Profit: the rate of change of profit with respect to the number of units sold. That is, P'(x).Marginal Profit: the rate of change of profit with respect to the number of units sold. That is, P'(x).

Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Change: Example The temperature during and after a thunderstorm can be modeled by T(h) = 2.37h 4 - 5.163h 3 + 8.69h 2 -9.87h + 78 degrees Fahrenheit where h is the number of hours after the storm began. Use the rate of change of T(h) at h = 0.25 to estimate how much the temperature changed between 15 and 20 minutes after the storm began. T'(.25) = -6.34 degrees per hour The temperature change is approximately (-6.34 °F/hour)(0.083 hours) = -0.53 °F. That is, the temperature dropped by about half a degree.

Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Change: Exercise 5.1 #13 The population of Mexico between 1921 and 1990 can be modeled by P(t) = 12.921e 0.026578t million people where t is the number of years since 1921. Estimate how much the population changed between 1985 and 1986. t = 64 in 1985 and t = 65 in 1986 P'(64) = 1.881698 million people per year P(65) - P(64)  P'(64) (65 - 64) = 1.881698 (1) = 1.881698 (1) = 1.881698 million people = 1.881698 million people

Copyright © by Houghton Mifflin Company, All rights reserved. Approximating Change: Exercise 5.1 #5 Interpret the following statements: a. At a production level of 500 units, marginal cost is \$17 per unit b. When weekly sales are 150 units, marginal profit is \$4.75 per unit a. Increasing production from 500 to 501 units will cost about \$17. b. Increasing sales from 150 to 151 units will increase profit by about \$4.75.

Copyright © by Houghton Mifflin Company, All rights reserved. Extrema Relative maximum: the largest output value relative to nearby outputsRelative maximum: the largest output value relative to nearby outputs Relative minimum: the smallest output value relative to nearby outputsRelative minimum: the smallest output value relative to nearby outputs Absolute maximum: the largest output value over the set of all inputs to the functionAbsolute maximum: the largest output value over the set of all inputs to the function Absolute minimum: the smallest output value over the set of all inputs to the functionAbsolute minimum: the smallest output value over the set of all inputs to the function

Copyright © by Houghton Mifflin Company, All rights reserved. Extrema If a function has a relative maximum or relative minimum at an input value c, then f '(c) = 0 or f '(c) does not exist but f(c) does exist.If a function has a relative maximum or relative minimum at an input value c, then f '(c) = 0 or f '(c) does not exist but f(c) does exist. If f '(c) =0 and the derivative crosses the input axis at c, then a relative extremum exists at c.If f '(c) =0 and the derivative crosses the input axis at c, then a relative extremum exists at c. An absolute extremum occurs at either a relative extremum or an endpoint of the specified interval.An absolute extremum occurs at either a relative extremum or an endpoint of the specified interval.

Copyright © by Houghton Mifflin Company, All rights reserved. Extrema: Example The population of Kentucky reached a relative maximum in 1983 and a relative minimum in 1988. The population absolute minimum occurred in 1980 and absolute maximum occurred in 1993.

Copyright © by Houghton Mifflin Company, All rights reserved. Extrema: Example B(x) models the number of consumer complaints about baggage on U.S. airlines between 1989 and 1997. A relative and absolute minimum occurs in 1994.

Copyright © by Houghton Mifflin Company, All rights reserved. Extrema: Exercise 5.2 #17 Sketch the graph of a function f that meets the following criteria: f has a relative minimum at x = 3 f has a relative maximum at x = -1 f '(x) > 0 for x 3 f '(x) < 0 for -1 < x < 3 f '(-1) = 0 and f '(-1) = 0

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization Without Equations 1 Identify the quantity to be maximized (output) or minimized and the quantity on which the output depends (the input). 2 Sketch and label a picture, if applicable. 3 Build a model for the quantity to be optimized. 4 Rewrite the model equation in terms of only one input variable, if necessary. 5 Identify the input interval. 6 Apply optimization techniques to the equation. 7 Answer the question posed in the problem.

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Example A rancher removed 200 feet of wire fencing from a field on his ranch. He wishes to reuse the fencing to create a rectangular corral into which he will build a 6-foot wide wooden gate. What dimensions will result in a corral with the greatest possible area? Step 1: output variable: area of rectangular corral input variables: length and width of corral

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Example Step 2: Step 3: A = lw where l is length, w is width, and A is area. This is the quantity we want to maximize.

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Example Step 4: The equation is Step 3 has two variables so we need to find a second equation that relates the length and width. 2w + 2l - 6 = 200 l = 103 - w So A = lw = (103 - w)w = 103w - w 2 square feet Step 5: If the corral length is 0 feet, the width is 100 feet. If the corral length is 100 feet, the width is 0 feet. So 0 < w < 100.

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Example Step 6: A'(w) = 103 - 2w Set A'(w) = 0 103 - 2w = 0 2w = 103 w = 51.5 feet will maximize corral area Step 7: l = 103 - w l = 103 - 51.5 = 51.5 feet So the corral should be 51.5 feet by 51.5 feet in order to maximize area.

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Exercise 5.3 #3 A rectangular-shaped garden has one side along the side of a house. The other three sides are to be enclosed with 60 feet of fencing. What is the largest possible area of the garden? Step 1: output variable: area of garden input variables: length and width of garden

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Exercise 5.3 #3 Step 2: Step 3: A = lw where l is length, w is width, and A is area. This is the quantity we want to maximize.

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Exercise 5.3 #3 Step 4: The equation is Step 3 has two variables so we need to find a second equation that relates the length and width. 2w + l = 60 l = 60 - 2w So A = lw = (60- 2w)w = 60w - 2w 2 square feet Step 5: If the garden length is 0 feet, the width is 30 feet. If the garden length is 60 feet, the width is 0 feet. So 0 < w < 30.

Copyright © by Houghton Mifflin Company, All rights reserved. Optimization: Exercise 5.3 #3 Step 6: A'(w) = 60 - 4w Set A'(w) = 0 60 - 4w = 0 4w = 60 w = 15 feet will maximize garden area Step 7: l = 60 - 2w l = 60 - 2(15) = 30 feet The garden should be 15 feet by 30 feet in order to maximize area. The long side is against the house.

Copyright © by Houghton Mifflin Company, All rights reserved. Inflection Points inflection point: a point where the graph changes concavityinflection point: a point where the graph changes concavity If c is an inflection point of a function f then f ''(c) = 0 or f ''(c) is undefinedIf c is an inflection point of a function f then f ''(c) = 0 or f ''(c) is undefined If the second derivative graph crosses the horizontal axis, then an inflection point of the function graph occurs at that input value.If the second derivative graph crosses the horizontal axis, then an inflection point of the function graph occurs at that input value.

Copyright © by Houghton Mifflin Company, All rights reserved. Inflection Points: Example The percentage of students graduating from high school in South Carolina from 1982 through 1990 who entered post- secondary institutions may be modeled by f(x) = -0.1057x 3 + 1.355x 2 - 3.672x + 50.792 percent where x is the number of years since 1982. Find the inflection point of the function. f ''(x) = 0 at x = 4.27 years since 1982

Copyright © by Houghton Mifflin Company, All rights reserved. Inflection Points: Exercise 5.4 #7 The graphs below present a function, its derivative, and its second derivative. Label the graphs and explain your reasoning. function; minimum at B inflection point at x = 0 inflection point at x = A second derivative; x-intercept at x =0 x-intercept at x =A derivative; x-intercept at B maximum at x = 0 minimum at x = A

Copyright © by Houghton Mifflin Company, All rights reserved. Related Rates 1 Identify independent and dependent variables 2 Find an equation relating the dependent variables. (Independent variable may or may not appear) 3 Differentiate both sides of the equation with respect to the independent variable 4 Substitute the known quantities and rates into the related-rates equation and solve for the unknown rate 5 Interpret the solution