SECTION 6. Partial Fractions and Logistic Growth 9 Section 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. 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 This method depends on the abilit to factor the denominator of the original rational function and on finding the partial fraction decomposition of the function. STUDY TIP Recall that finding the partial fraction decomposition of a rational function is 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.
CHAPTER 6 Techniques of Integration Eample Finding a Partial Fraction Decomposition Algebra Review You can check the result in Eample b subtracting the partial fractions to obtain the original fraction, as shown in Eample (a) in the Chapter 6 Algebra Review, on page 7. Write the partial fraction decomposition for 7 6. 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 particular factors equal to. To solve for B, substitute : 7 A B 7 A B A B B To solve for A, 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. CHECKPOINT 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.
SECTION 6. Partial Fractions and Logistic Growth TECHNOLOGY 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 (b) in the Chapter 6 Algebra Review, on page 7. Also, for help with the algebra used to simplif the answer, see Eample (c) on page 7. Eample Integrating with Repeated Factors 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 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, 6 A 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 CHECKPOINT Find 7 d.
CHAPTER 6 Techniques of Integration 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. 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. Eample Find 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. A B C D 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. d d d ln C CHECKPOINT Find.
Logistic growth model: growth is restricted. FIGURE 6. = L t SECTION 6. Partial Fractions and Logistic Growth 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 ddt k. The general solution of this differential equation is Ce kt. 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.. This upper limit L is called the carring capacit, which is the maimum population t that can be sustained or supported as time t increases. A model that is often used for this tpe of growth is the logistic differential equation d dt k L Logistic differential equation where k and L are positive constants. A population that satisfies this equation does not grow without bound, but approaches L as t increases. The general solution of this differential equation is called the logistic growth model and is derived in Eample. STUDY TIP The graph of L be kt is called the logistic curve, as shown in Figure 6.. Algebra Review For help with the algebra used to solve for in Eample, see Eample (c) in the Chapter 6 Algebra Review, on page 7. CHECKPOINT Show that if then be kt, d k. dt [Hint: First find k in terms of t, then find ddt and show that the are equivalent.] Eample Solve the equation SOLUTION Deriving the Logistic Growth Model d dt k L. d k dt L L d k dt L d k dt ln ln L kt C ln L kt C L ektc e C e kt L d dt k L be kt Write differential equation. Write in differential form. Integrate each side. Rewrite left side using partial fractions. Find antiderivative of each side. Multipl each side b and simplif. Eponentiate each side. Let ±e C b. Solving this equation for L produces the logistic growth model be. kt
CHAPTER 6 Techniques of Integration Eample 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.). = + e t = + e t = + e t FIGURE 6. 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.). = +.e t = + e t = + e t FIGURE 6. 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 6.6). = + e.t = + e t = + e t FIGURE 6.6 CHECKPOINT Find the horizontal asmptote of the graph of e 6t.
SECTION 6. Partial Fractions and Logistic Growth Eample 6 Modeling a Population CHECKPOINT 6 Daniel J. Co/Gett Images Write the logistic growth model for the population of deer in Eample 6 if the game preserve could contain a limit of deer. 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 a 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 dt 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, 8 766 9 99 998 CONCEPT CHECK. Complete the following: The technique of partial fractions involves the decomposition of a function into the of two or more simple functions.. What is a proper rational function?. Before appling partial fractions to an improper rational function, what should ou do?. Describe what the value of L represents in the logistic growth function L be kt.
6 CHAPTER 6 Techniques of Integration Skills 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. For additional help, review Sections. and.. In Eercises 8, factor the epression.. 6.... 6. 7. 8. 6 7 In Eercises 9, rewrite the improper rational epression as the sum of a proper rational epression and a polnomial. 9...... Eercises 6. In Eercises, write the partial fraction decomposition for the epression.... 8.. 6. 7. 8. 9.. 8. 9. 7 6 6 In Eercises, use partial fractions to find the indefinite integral... d d. 6. d 6 d 7. 8. d d 9.. 6 d d.. 9 d d.. d d See www.calcchat.com for worked-out solutions to odd-numbered eercises.. d 6. 7. 8. d 9.. d.. d In Eercises, evaluate the definite integral... d 9 d. 6. d d 7. 8. d 9. d. In Eercises, find the area of the shaded region... 6 6 7 6 d 6 9 d d d d d
SECTION 6. Partial Fractions and Logistic Growth 7.. In Eercises and 6, find the area of the region bounded b the graphs of the given equations.. 6. In Eercises 7, 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. 7. a 8. 9. a.. Writing What is the first step when integrating Eplain. (Do not integrate.) d?. Writing State the method ou would use to evaluate each integral. Eplain wh ou chose that method. (Do not integrate.) 7 (a) (b) 8 d 8 d. Biolog A conservation organization releases animals of an endangered species into a game preserve. During the first ears, the population increases to animals. 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 dt,,, 6 a a 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 dt 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. (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) 9 8 7 6 Bacterial Culture (, ) 6 8 Time (in hours) t
8 CHAPTER 6 Techniques of Integration 7. Revenue The revenue R (in millions of dollars per ear) for Smantec Corporation from 997 through can be modeled b R t,t,7 6t 9t where t 7 corresponds to 997. Find the total revenue from 997 through. Then find the average revenue during this time period. (Source: Smantec Corporation) 8. Environment The predicted cost C (in hundreds of thousands of dollars) for a compan to remove p% of a chemical from its waste water is shown in the table. p C.7...7 p 6 7 8 9 C..7.6.. A model for the data is given b p C p p, p <. Use the model to find the average cost for removing between 7% and 8% of the chemical. 9. 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 Paramecium Population P. aurelia P. caudatum 6 8 6 Das 6. Population Growth The population of the United States was 76 million people in 9 and reached million people in 6. From 9 through 6, assume the population of the United States can be modeled b logistic growth with a limit of 89. million people. (Source: U.S. Census Bureau) (a) Write a differential equation of the form d dt k L where represents the population of the United States (in millions of people) and t represents the number of ears since 9. L (b) Find the logistic growth model for this be kt population. (c) Use a graphing utilit to graph the model from part (b). Then estimate the ear in which the population of the United States will reach million people. Business Capsule Photo courtes of Susie Wang and Ric Kostick Susie Wang and Ric Kostick graduated from the Universit of California at Berkele with degrees in mathematics. In 999, Wang used $, to start Aqua Dessa Spa Therap, a high-end cosmetics compan that uses natural ingredients in their products. Now, the compan run b Wang and Kostick has annual sales of over $ million, operates under several brand names, including % Pure, and has a global customer base. Wang and Kostick attribute the success of their business to appling what the learned from their studies. 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.
A Answers to Selected Eercises. u ; dv e d. u ln ; dv d. e 9 e C 7. e e e C 9. ln C. e C. e 6 e C. e C 7. e e C 9. e e e C. t lnt ln t t t C. e e C. e t C 7. ln ln C 9. ln C. ln C. C. e 7. C C 9. ee.6. e.76. 6 e6 6 6.6. ln.86 7. Area e 6.778 6 9. Area 9e.7 8. Proof.. ln C 7. e.98 9. e.97 6. 8 79 8 e8. 6. 6.,7,87 6,, e C,8.7 (a) Increase (b), units (c), unitsr 67. (a). ln..8 (b).8 ln 7. ln.8 8. 69. $8,8. 7. $9,6. 7. $.7 7. (a) $,, (b) $,9,.7 77. $,97.78 79. (a) $7,78.6 (b) $68.6 8.. SECTION 6. (page 6) Skills Review (page 6)..... 6. 7. 8. 9....... 9.. 7. 9. 8... 7. ln C 9.. ln C C. ln ln C. 7. ln C 9.. 6,, ln C ln ln C ln C ln ln C.. 6 ln 7.9. 7. ln.9 9. ln.7. 7 ln 7.89. ln ln.86
Answers to Selected Eercises A. ln 6 ln. 7. 9. a a a a a. Divide b because the degree of the numerator is greater than the degree of the denominator.. 9e.66t. $.77 thousand 7. $,8 million; $6 million 9. The rate of growth is increasing on, for P. aurelia and on, for P. caudatum; the rate of growth is decreasing on, for P. aurelia and on, for P. caudatum; P. aurelia has a higher limiting population. 6. Answers will var.... 7. 9. C... 7. 7 t ln s s C ln C 8 7 ln C 9 ln ln ln C Area 9 C t ln t C 9 9 ln C SECTION 6. (page 7) Skills Review (page 7). 8 6... 9. 6. 7. 8. 9. e C. ln C 9.. Area. Area ln.. Area.7 Area 8 ln 8 ln e. 9 ln C. C. ln 7. 9. ln 9 C 7 e C C. ln 9 C. ln C. ln C 7. ln e C 9. ln C Area 9.8..9. 9 7. ln ln 9. e 6.796 9. 8 ln 7.... (a).8 (b).8 ln e C C