6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH

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1 6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life situations. Partial Fractions In Sections 6. and 6., ou studied integration b substitution and b parts. In this section ou will stud a third technique called partial fractions. This technique involves the decomposition of a rational function into the sum of two or more simple rational functions. For instance, suppose ou know that 7 6. Knowing the partial fractions on the right side would allow ou to integrate the left side as shown. 7 6 d d d d ln ln C To use this method, ou must be able to factor the denominator of the original rational function and find the partial fraction decomposition of the function. STUDY TIP Finding the partial fraction decomposition of a rational function is reall a precalculus topic. Eplain how ou could verif that is the partial fraction decomposition of. Partial Fractions To find the partial fraction decomposition of the proper rational function pq, factor q and write an equation that has the form p sum of partial fractions). q For each distinct linear factor a b, the right side should include a term of the form A a b. For each repeated linear factor a b n, the right side should include n terms of the form A a b A a b... A n a b. n STUDY TIP A rational function pq is proper if the degree of the numerator is less than the degree of the denominator.

2 SECTION 6. Partial Fractions and Logistic Growth 7 EXAMPLE Write the partial fraction decomposition for 7 6. Finding a Partial Fraction Decomposition SOLUTION Begin b factoring the denominator as 6. Then, write the partial fraction decomposition as 7 6 To solve this equation for A and B,multipl each side of the equation b the least common denominator. This produces the basic equation as shown. 7 A B Basic equation Because this equation is true for all, ou can substitute an convenient values of into the equation. The -values that are especiall convenient are the ones that make a factor of the least common denominator zero: and. Substitute : 7 A B 7 A B A B B Substitute : 7 A B 7 A B A B A A B. Write basic equation. Substitute for. Simplif. Solve for B. Write basic equation. Substitute for. Simplif. Solve for A. Now that ou have solved the basic equation for A and B,ou can write the partial fraction decomposition as 7 6 as indicated at the beginning of this section. ALGEBRA REVIEW You can check the result in Eample b subtracting the partial fractions to obtain the original fraction, as shown in Eample (b) in the Chapter 6 Algebra Review, on page 6. TRY IT Write the partial fraction decomposition for 8 7. STUDY TIP Be sure ou see that the substitutions for in Eample are chosen for their convenience in solving for A and B. The value is selected because it eliminates the term A, and the value is chosen because it eliminates the term B.

3 8 CHAPTER 6 Techniques of Integration TECHNOLOGY EXAMPLE Integrating with Repeated Factors The use of partial fractions depends on the abilit to factor the denominator. If this cannot be easil done, then partial fractions should not be used. For instance, consider the integral 6 d. This integral is onl slightl different from that in Eample, et it is immensel more difficult to solve. A smbolic integration utilit was unable to solve this integral. Of course, if the integral is a definite integral (as is true in man applied problems), then ou can use an approimation technique such as the Midpoint Rule. ALGEBRA REVIEW You can check the partial fraction decomposition in Eample b combining the partial fractions to obtain the original fraction, as shown in Eample (c) in the Chapter 6 Algebra Review, on page 6. Also, for help with the algebra used to simplif the answer, see Eample (c) on page 7. Find 6 d. SOLUTION Begin b factoring the denominator as. Then, write the partial fraction decomposition as 6 A B C. To solve this equation for A, B, and C, multipl each side of the equation b the least common denominator. 6 A B C Basic equation Now, solve for A and C b substituting and into the basic equation. Substitute : 6 A B C Substitute : 6 A B C 6 A B C 6 A Solve for C. Solve for A. At this point, ou have ehausted the convenient choices for and have et to solve for B. When this happens, ou can use an other -value along with the known values of A and C. Substitute, A 6, and C 9: 6 6 B 9 6 B 9 B 9 A B C 9 C Solve for B. Now that ou have solved for A, B, and C,ou can use the partial fraction decomposition to integrate. 6 d 6 9 d 9 C ln 6 9 C 6 ln ln TRY IT Find 7 d.

4 You can use the partial fraction decomposition technique outlined in Eamples and onl with a proper rational function that is, a rational function whose numerator is of lower degree than its denominator. If the numerator is of equal or greater degree, ou must divide first. For instance, the rational function is improper because the degree of the numerator is greater than the degree of the denominator. Before appling partial fractions to this function, ou should divide the denominator into the numerator to obtain. SECTION 6. Partial Fractions and Logistic Growth 9 Find EXAMPLE Integrating an Improper Rational Function SOLUTION This rational function is improper its numerator has a degree greater than that of its denominator. So, ou should begin b dividing the denominator into the numerator to obtain Now, appling partial fraction decomposition produces Multipling both sides b the least common denominator produces the basic equation. Basic equation Using techniques similar to those in the first two eamples, ou can solve for A, B, C, and D to obtain A, B, C, and D. So, ou can integrate as shown. d.. A B C D A B C D. d d d ln C ALGEBRA REVIEW You can check the partial fraction decomposition in Eample b combining the partial fractions to obtain the original fraction, as shown in Eample (a) in the Chapter 6 Algebra Review, on page 7. TRY IT Find.

5 CHAPTER 6 Techniques of Integration = L Logistic growth model: growth is restricted. FIGURE 6.6 t Logistic Growth Function In Section.6, ou saw that eponential growth occurs in situations for which the rate of growth is proportional to the quantit present at an given time. That is, if is the quantit at time t, then d k Ce kt. d is proportional to. Eponential growth function Eponential growth is unlimited. As long as C and k are positive, the value of Ce kt can be made arbitraril large b choosing sufficientl large values of t. In man real-life situations, however, the growth of a quantit is limited and cannot increase beond a certain size L, as shown in Figure 6.6. The logistic growth model assumes that the rate of growth is proportional to both the quantit and the difference between the quantit and the limit L. That is d kl. d is proportional to and L. The solution of this differential equation is given in Eample. STUDY TIP The logistic growth model in Eample is simplified b assuming that the limit of the quantit is. If the limit were L, then the solution would be In the fourth step of the solution, notice that partial fractions are used to integrate the left side of the equation. TRY IT Show that if L be kt. then be kt, d k. [Hint: First find k in terms of t, then find d and show that the are equivalent.] EXAMPLE Solve the equation d k. Assume > and >. SOLUTION d k d k ln ln kt C ln kt C Solving this equation for produces be kt where b C. d k Deriving the Logistic Growth Function d k Cekt Write differential equation. Write in differential form. Integrate each side. Rewrite using partial fractions. Find antiderivative. Simplif. Eponentiate and let e C C. Logistic growth function

6 SECTION 6. Partial Fractions and Logistic Growth EXAMPLE Comparing Logistic Growth Functions Use a graphing utilit to investigate the effects of the values of L, b, and k on the graph of L be kt. Logistic growth function L >, b >, k > SOLUTION The value of L determines the horizontal asmptote of the graph to the right. In other words, as t increases without bound, the graph approaches a limit of L (see Figure 6.7). = + e t = + e t = + e t FIGURE 6.7 The value of b determines the point of inflection of the graph. When b, the point of inflection occurs when t. If b >, the point of inflection is to the right of the -ais. If < b <, the point of inflection is to the left of the -ais (see Figure 6.8). = +.e t = + e t = + e t FIGURE 6.8 The value of k determines the rate of growth of the graph. For fied values of b and L, larger values of k correspond to higher rates of growth (see Figure 6.9). = + e.t = + e t = + e t FIGURE 6.9 TRY IT Find the horizontal asmptote of the graph of e 6t.

7 CHAPTER 6 Techniques of Integration Galen Rowell/CORBIS The American peregrine falcon was removed from the endangered species list in 999 due to its recover from nesting pairs in North America in 97 to 6 pairs in the United States and Canada. The peregrine was put on the endangered species list in 97 because of the use of the chemical pesticide DDT. The Fish and Wildlife Service, state wildlife agencies, and man other organizations contributed to the recover b setting up protective breeding programs among other efforts. EXAMPLE 6 Modeling a Population The state game commission releases deer into a game preserve. During the first ears, the population increases to deer. The commission believes that the population can be modeled b logistic growth with a limit of deer. Write the logistic growth model for this population. Then use the model to create a table showing the size of the deer population over the net ears. SOLUTION Let represent the number of deer in ear t. Assuming a logistic growth model means that the rate of change in the population is proportional to both and. That is d k, The solution of this equation is be kt. Using the fact that when t, ou can solve for b. be k Then, using the fact that when t, ou can solve for k. So, the logistic growth model for the population is 9e k 9e.6t.. b 9 k.6 Logistic growth model The population, in five-ear intervals, is shown in the table. Time, t Population, TRY IT 6 Write the logistic growth model for the population of deer in Eample 6 if the game preserve could contain a limit of deer. T AKE ANOTHER LOOK Logistic Growth Analze the graph of the logistic growth function in Eample 6. During which ears is the rate of growth of the herd increasing? During which ears is the rate of growth of the herd decreasing? How would these answers change if, instead of a limit of deer, the game preserve could contain a limit of deer?

8 SECTION 6. Partial Fractions and Logistic Growth PREREQUISITE REVIEW 6. The following warm-up eercises involve skills that were covered in earlier sections. You will use these skills in the eercise set for this section. In Eercises 8, factor the epression In Eercises 9, rewrite the improper rational epression as the sum of a proper rational epression and a polnomial EXERCISES 6. In Eercises, write the partial fraction decomposition for the epression In Eercises, find the indefinite integral d d. 6. d 6 d d d 9.. d.. d.. d. d d 9.. d.. d In Eercises, evaluate the definite integral d d d d d 6 d 9 d d 7 d d 6 9 d

9 CHAPTER 6 Techniques of Integration In Eercises, find the area of the shaded region In Eercises 8, write the partial fraction decomposition for the rational epression. Check our result algebraicall. Then assign a value to the constant a and use a graphing utilit to check the result graphicall.. a a 8. In Eercises 9, use a graphing utilit to graph the function. Then find the volume of the solid generated b revolving the region bounded b the graphs of the given equations about the -ais b using the integration capabilities of a graphing utilit and b integrating b hand using partial fraction decomposition. 9. d d d d 7 6,,, a a...,. Biolog A conservation organization releases animals of an endangered species into a game preserve. The organization believes that the preserve has a capacit of animals and that the herd will grow according to a logistic growth model. That is, the size of the herd will follow the equation,, 6,,,, d k,, where t is measured in ears. Find this logistic curve. (To solve for the constant of integration C and the proportionalit constant k, assume when t and when t. ) Use a graphing utilit to graph our solution.. Health: Epidemic A single infected individual enters a communit of individuals susceptible to the disease. The disease spreads at a rate proportional to the product of the total number infected and the number of susceptible individuals not et infected. A model for the time it takes for the disease to spread to individuals is t d where t is the time in hours. (a) Find the time it takes for 7% of the population to become infected (when t, ). (b) Find the number of people infected after hours.. Marketing After test-marketing a new menu item, a fast-food restaurant predicts that sales of the new item will grow according to the model ds t t where t is the time in weeks and S is the sales (in thousands of dollars). Find the sales of the menu item at weeks. 6. Biolog One gram of a bacterial culture is present at time t, and grams is the upper limit of the culture s weight. The time required for the culture to grow to grams is modeled b kt d where is the weight of the culture (in grams) and t is the time in hours.

10 SECTION 6. Partial Fractions and Logistic Growth (a) Verif that the weight of the culture at time t is modeled b 9e. kt Use the fact that when t. (b) Use the graph to determine the constant k. Weight (in grams) Bacterial Culture 7. Revenue The revenue R (in millions of dollars per ear) for Smantec Corporation from 99 through can be modeled b R t 8,9t 8,8 6t 9t where t corrresponds to 99. Find the total revenue from 99 through. Then find the average revenue during this time period. (Source: Smantec Corporation) 8. Medicine On a college campus, students return from semester break with a contagious flu virus. The virus has a histor of spreading at a rate of dn e.t e.t where N is the number of students infected after t das. (a) Find the model giving the number of students infected with the virus in terms of the number of das since returning from semester break. (b) If nothing is done to stop the virus from spreading, will the virus spread to infect half the student population of students? Eplain our answer. 9. Biolog A conservation organization releases animals of an endangered species into a game preserve. The organization believes the population of the species will increase at a rate of dn e.t 9e.t (, ) 6 8 Time (in hours) where N is the population and t is the time in months. (a) Use the fact that N when t to find the population after ears. (b) Find the limiting size of the population as time increases without bound. t 6. Biolog: Population Growth The graph shows the logistic growth curves for two species of the single-celled Paramecium in a laborator culture. During which time intervals is the rate of growth of each species increasing? During which time intervals is the rate of growth of each species decreasing? Which species has a higher limiting population under these conditions? (Source: Adapted from Levine/Miller, Biolog: Discovering Life, Second Edition) Number Courtes of Susie Wang/Aqua Dessa Paramecium Population P. aurelia P. caudatum Das BUSINESS CAPSULE While a math communications major at the Universit of California at Berkele, Susie Wang began researching the idea of selling natural skin-care products. She used $, to start her compan, Aqua Dessa, and uses word-of-mouth as an advertising tactic. Aqua Dessa products are used and sold at spas and eclusive cosmetics counters throughout the United States. 6. Research Project Use our school s librar, the Internet, or some other reference source to research the opportunit cost of attending graduate school for ears to receive a Masters of Business Administration (MBA) degree rather than working for ears with a bachelor s degree. Write a short paper describing these costs.