Rates of Change in Rational Functions. LEARN ABOUT the Math

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1 . Rates of Change in Rational Functions YOU WILL NEED graphing calculator or graphing software GOAL Determine average rates of change, and estimate instantaneous rates of change for rational functions. LEARN ABOUT the Math The instantaneous rate of change at a point on a revenue function is called the marginal revenue. It is a measure of the estimated additional revenue from selling one more item. For eample, the demand equation for a toothbrush is p(), where is the number of toothbrushes sold, in thousands, and p is the price, in dollars.? What is the marginal revenue when toothbrushes are sold? When is the marginal revenue the greatest? When is it the least? EXAMPLE Selecting a strateg to determine instantaneous rates of change Determine the marginal revenue when toothbrushes are sold and when it is the greatest and the least. Solution A: Calculating the average rate of change b squeezing centred intervals around. Revenue R() p() Revenue Number of items sold 3 Price The average rate of change close to. is shown in the following table. Centred Intervals Average Rate of Change R( ) R( ). # #..7. # #...9 # #...99 # #.. is measured in thousands, so when toothbrushes are sold,.. The average rate of change from (, R( )) to (, R( )) is the slope of the secant that joins each pair of endpoints. 9. Rates of Change in Rational Functions NEL

2 The average rate of change approaches.. The marginal revenue when toothbrushes are sold is $. per toothbrush. Sketch the graph of R(). The graph starts at (, ) and has a horizontal asmptote at. = R() = + = The marginal revenue is the greatest when and then decreases from there, approaching zero, as the graph gets closer to the horizontal asmptote. When and are ver close to each other, the slope of the secant is approimatel the same as the slope of the tangent.the slopes of the secants near the point where. approach.. The vertical asmptote at is not in the domain of R(), since $, so it can be ignored. In the contet of the problem, and R() have onl positive values. Eamine the slope of the tangent lines at various points along the domain of the revenue graph. The slope is the greatest at the beginning of the graph and then decreases as increases.. Solution B: Using the difference quotient and graphing technolog to analze the revenue function Enter the revenue function into a graphing calculator. Average rate of change Let h. R(a h) R(a) h R(..) R(.). R(.) R(.). Use the difference quotient and a ver small value for h, where a., to estimate the instantaneous rate of change in revenue when toothbrushes are sold. NEL Chapter 99

3 Enter the rate of change epression into the graphing calculator to determine its value, using the equation entered into Y. The average rate of change is about.. The marginal revenue when toothbrushes are sold is $. per toothbrush. To verif, graph the revenue function R() p() and draw a tangent line at.. Since and R() onl have positive values, graph the function in the first quadrant. When toothbrushes are sold, the marginal revenue is $. per toothbrush. Use the DRAW feature of the graphing calculator to draw a tangent line where.. The marginal revenue is the greatest when. The marginal revenue decreases to ver small values as increases. The tangent lines to this curve are steepest at the beginning of the curve. Their slopes decrease as increases. Reflecting A. In Solution A, how were average rates of change used to estimate the instantaneous rate of change at a point? B. In Solutions A and B, how were graphs used to estimate the instantaneous rate of change at a point? C. In each solution, how was it determined where the marginal revenue was the greatest? Wh was it not possible to determine the least marginal revenue? 3. Rates of Change in Rational Functions D. What are the advantages and disadvantages of each method to determine the instantaneous rate of change? NEL

4 APPLY the Math. EXAMPLE a) Estimate the slope of the tangent to the graph of f () at the point where. 3 Wh can there not be a tangent line where 3? Solution Connecting the instantaneous rate of change to the slope of a tangent a) f () 3 average rate of change Let h. f (a h) f (a) h f (.) f (). f (.999) f ()..999 a b a 3 b Use the difference quotient and a ver small value for h close to a to estimate the slope of the tangent where. The slope of the tangent at is.7. The value 3 is not in the domain of f (), so no tangent line is possible there. The graph of f () has a vertical asmptote at 3. As approaches 3 from the left and from the right, the tangent lines are ver steep. The tangent lines approach a vertical line, but are never actuall vertical. There is no point on the graph with an -coordinate of 3, so there is no tangent line there. NEL Chapter 3

5 EXAMPLE 3 Selecting a graphing strateg to solve a problem that involves average and instantaneous rates of change The snowshoe hare population in a newl created conservation area can be predicted over time b the model p(t) t where p represents the population size and t is the time in ears since the opening of the t, conservation area. Determine when the hare population will increase most rapidl, and estimate the instantaneous rate of change in population at this time. Solution Graph p(t) t for # t #. t p(t) $, but ou can set the minimum -value to a negative number to allow some space for displaed values. The slopes of the tangent lines increase slowl at the beginning of the graph. The slopes start to increase more rapidl around t. The begin to decrease after t 3. Draw tangent lines between and 3, and look for the tangent line that has the greatest slope. The slopes of the tangent lines increase until.9, and then decrease. The tangent line at t.9 has the greatest slope. The average rate of change is greatest when t is close to.9. The hare population will increase most rapidl about ears and months after the conservation area is opened. The instantaneous rate of change in population at this time is approimatel 3 hares per ear..9 3 months is approimatel months. 3. Rates of Change in Rational Functions NEL

6 . In Summar Ke Ideas The methods that were previousl used to calculate the average rate of change and estimate the instantaneous rate of change can be used for rational functions. You cannot determine the average and instantaneous rates of change of a rational function at a point where the graph is discontinuous (that is, where there is a hole or a vertical asmptote). Need to Know The average rate of change of a rational function, f(), on the interval f( from is ) f( ) # #. Graphicall, this is equivalent to the slope of the secant line that passes through the points (, ) and (, ) on the graph of f(). The instantaneous rate of change of a rational function, f(), at a can f(a h) f(a) be approimated using the difference quotient and a ver small h value of h. Graphicall, this is equivalent to estimating the slope of the tangent line that passes through the point (a, f(a)) on the graph of f(). The instantaneous rate of change at a vertical asmptote is undefined. The instantaneous rates of change at points that are approaching a vertical asmptote become ver large positive or ver large negative values. The instantaneous rate of change near a horizontal asmptote approaches zero. CHECK Your Understanding. The graph of a rational function is shown. a) Determine the average rate of change of the function over the interval # # 7. Cop the graph, and draw a tangent line at the point where. Determine the slope of the tangent line to estimate the instantaneous rate of change at this point.. Estimate the instantaneous rate of change of the function in question at b determining the slope of a secant line from the point where to the point where.. Compare our answer with our answer for question, part. = Use graphing technolog to estimate the instantaneous rate of change of the function in question at. PRACTISING. Estimate the instantaneous rate of change of f () at the point (, ). NEL Chapter 33

7 . Select a strateg to estimate the instantaneous rate of change of each K function at the given point. a), where 3 7 3, where 3, where 3, where 3. Determine the slope of the line that is tangent to the graph of each function at the given point. Then determine the value of at which there is no tangent line. a) f () 3, where f (), where 7 f (), where 3 f (), where 7. When polluted water begins to flow into an unpolluted pond, the A concentration of pollutant, c, in the pond at t minutes is modelled b 7t c(t) where c is measured in kilograms per cubic metre. 3t, Determine the rate at which the concentration is changing after a) h one week. The demand function for snack cakes at a large baker is given b the function p() The -units are given in thousands of. cakes, and the price per snack cake, p(), is in dollars. a) Find the revenue function for the cakes. Estimate the marginal revenue for.7. What is the marginal revenue for.? 9. At a small clothing compan, the estimated average cost function for producing a new line of T-shirts is C(), where is the number of T-shirts produced, in thousands. C() is measured in dollars. a) Calculate the average cost of a T-shirt at a production level of 3 pairs. Estimate the rate at which the average cost is changing at a production level of 3 T-shirts. 3. Rates of Change in Rational Functions NEL

8 . Suppose that the number of houses in a new subdivision after t months of development is modelled b N(t) t 3, where t 3 N is the number of houses and # t #. a) Calculate the average rate of change in the number of houses built over the first months. Calculate the instantaneous rate of change in the number of houses built at the end of the first ear. Graph the function using a graphing calculator. Discuss what happens to the rate at which houses were built in this subdivision during the first ear of development.. Given the function f () determine an interval and a, T point where the average rate of change and the instantaneous rate of change are equal... a) The position of an object that is moving along a straight line at C t seconds is given b s(t) 3t where s is measured in metres. t, Eplain how ou would determine the average rate of change of s(t) over the first s. What does the average rate of change mean in this contet? Compare two was that ou could determine the instantaneous rate of change when t. Which method is easier? Eplain. Which method is more accurate? Eplain. What does the instantaneous rate of change mean in this contet? Etending 3. The graph of the rational function f () has been given the name Newton s Serpentine. Determine the equations for the tangents at the points where "3,, and "3.. Determine the instantaneous rate of change of Newton s Serpentine at points around the point (, ). Then determine the instantaneous rate of change of this instantaneous rate of change. NEL Chapter 3

9 Chapter Review Stud Aid See Lesson., Eamples,, and 3. Tr Chapter Review Questions 7,, and 9. FREQUENTLY ASKED Questions Q: How do ou solve and verif a rational equation such as 3? A: You can solve a simple rational equation algebraicall b multipling each term in the equation b the lowest common denominator and then solving the resulting polnomial equation. 3 For eample, to solve multipl the equation b, ( )( ), where or Then solve the resulting. polnomial equation. To verif our solutions, ou can graph the corresponding function, f () 3 using graphing technolog and determine, the zeros of f. Stud Aid See Lesson., Eamples,, and 3. Tr Chapter Review Questions and. The zeros are and, so the solution to the equation is or. Q: How do ou solve a rational inequalit, such as? > A: You can solve a rational inequalit algebraicall b creating and solving an equivalent linear or polnomial inequalit with zero on one side. For factorable polnomial inequalities of degree or more, use a table to identif the positive/negative intervals created b the zeros and vertical asmptotes of the rational epression. A: You can use graphing technolog to graph the functions on both sides of the inequalit, determine their intersection and the locations of all vertical asmptotes, and then note the intervals of that satisf the inequalit. 3 Chapter Review NEL

10 Chapter Review Q: How do ou determine the average or instantaneous rate of change of a rational function? A: You can determine average and instantaneous rates of change of a rational function at points within the domain of the function using the same methods that are used for polnomial functions. Stud Aid See Lesson., Eamples,, and 3. Tr Chapter Review Questions, 3, and. Q: When is it not possible to determine the average or instantaneous rate of change of a rational function? A: You cannot determine the average and instantaneous rates of change of a rational function at a point where the graph has a hole or a vertical asmptote. You can onl calculate the instantaneous rate of change at a point where the rational function is defined and where a tangent line can be drawn. A rational function is not defined at a point where there is a hole or a vertical asmptote. For eample, f () and 3 defined at 3. g () 9 3 are rational functions that are not f() = + 3 g() = 9 3 = 3 The graph of f () has a The graph of g() has a hole vertical asmptote at 3. at 3. You cannot draw a tangent line on either graph at 3, so ou cannot determine an instantaneous rate of change at this point. NEL Chapter 37

11 PRACTICE Questions Lesson.. For each function, determine the domain and range, intercepts, positive/negative intervals, and increasing and decreasing intervals. Use this information to sketch a graph of the reciprocal function. a) f () 3 f () 7 f (). Given the graphs of f () below, sketch the graphs of a) f (). = f () = f () Lesson. 3. For each function, determine the equations of an vertical asmptotes, the locations of an holes, and the eistence of an horizontal or oblique asmptotes. a) 7 Lesson.3. The population of locusts in a Prairie a town over the last ears is modelled b the function 7 f () 3. The locust population is given in hundreds of thousands. Describe the locust population in the town over time, where is time in ears.. For each function, determine the domain, intercepts, asmptotes, and positive/negative intervals. Use these characteristics to sketch the graph of the function. Then describe where the function is increasing or decreasing. a) f () f () f () 3 f () 3. Describe how ou can determine the behaviour of the values of a rational function on either side of a vertical asmptote. 3 Chapter Review NEL

12 Lesson. 7. Solve each equation algebraicall, and verif our solution using a graphing calculator. a). A group of students have volunteered for the student council car wash. Janet can wash a car in m minutes. Rodriguez can wash a car in m minutes, while Nick needs the same amount of time as Janet. If the all work together, the can wash a car in about 3.3 minutes. How long does Janet take to wash a car? 9. The concentration of a toic chemical in a spring-fed lake is given b the equation c() where c is given in grams 3, per litre and is the time in das. Determine when the concentration of the chemical is. g> L. Lesson.. Use an algebraic process to find the solution set of each inequalit. Verif our answers using graphing technolog. a), #. A biologist predicted that the population of tadpoles in a pond could be modelled b the function f (t) t where t is given in t, das. The function that actuall models the t tadpole population is g(t). t t 7 Determine where g(t). f (t). Lesson.. Estimate the slope of the line that is tangent to each function at the given point. At what point(s) is it not possible to draw a tangent line? a) f () 3 3, where f () 3, where Chapter Review 3. The concentration, c, of a drug in the bloodstream t hours after the drug was taken orall is given b c(t) t where c is t 7, measured in milligrams per litre. a) Calculate the average rate of change in the drug s concentration during the first h since ingestion. Estimate the rate at which the concentration of the drug is changing after eactl 3 h. Graph c(t) on a graphing calculator. When is the concentration of the drug increasing the fastest in the bloodstream? Eplain.. Given the function f () determine, the coordinates of a point on f () where the slope of the tangent line equals the slope of the secant line that passes through A(, ) and B(, ).. Describe what happens to the slope of a tangent line on the graph of a rational function as the -coordinate of the point of tangenc a) gets closer and closer to the vertical asmptote. grows larger in both the positive and negative direction. NEL Chapter 39

13 Chapter Review Self-Test. Match each graph with the equation of its corresponding function. a) = = A B. Suppose that n is a constant and that f () is a linear or quadratic function defined when n. Complete the following sentences. a) If f (n) is large, then is... f (n) If f (n) is small, then is... f (n) If f (n), then is... f (n) If f (n) is positive, then is... f (n) 3. Without using graphing technolog, sketch the graph of.. A compan purchases kilograms of steel for $9.. The compan processes the steel and turns it into parts that can be used in other factories. After this process, the total mass of the steel has dropped b kg (due to trimmings, scrap, and so on), but the value of the steel has increased to $ 3.. The compan has made a profit of $> kg. What was the original mass of the steel? What is the original cost per kilogram?. Select a strateg to solve each of the following. a) If ou are given the equation of a rational function of the form f () a b eplain c d, a) how ou can determine the equations of all vertical and horizontal asmptotes without graphing the function when this tpe of function would have a hole instead of a vertical asmptote = 3 Chapter Self-Test NEL

14 Chapter Task A New School Researchers at a school board have developed models to predict population changes in the three areas the service. The models are A(t) 3 t for area A, B(t) 3t for area B, and C(t) for area C, t t where the population is measured in thousands and t is the time, in ears, since 7. The eisting schools are full, and the board has agreed that a new school should be built.? In which area should the new school be built, and when will the new school be needed? A. Graph each population function for the ears following 7. Use our graphs to describe the population trends in each area between 7 and 7. B. Describe the intervals of increase or decrease for each function. C. Determine which area will have the greatest population in, 7,, and 7. D. Determine the intervals over which the population of area A is greater than the population of area B the population of area A is greater than the population of area C the population of area B is greater than the population of area C E. Determine when the population of area B will be increasing most rapidl and when the population of area C will be increasing most rapidl. F. What will happen to the population in each area over time? G. Decide where and when the school should be built. Compile our results into a recommendation letter to the school board. Task Checklist Did ou show all our steps? Did ou draw and label our graphs accuratel? Did ou support our choice of location for the school? Did ou eplain our thinking clearl? NEL Chapter 3

15 . a) Answers ma var. For eample,. 7 Answers ma var. For eample, 3 $ Answers ma var. For eample, 3 # Answers ma var. For eample, 9,,3 7. a) Pa, `b P c 3, `b P(`, ) P(`, 3. a) PR,, PR # # PR 3 # # PR,, 9. a) The second plan is better if one calls more than 3 min per month a),, # 3 or $, or,, 7 #or # #. negative when P(, ), positive when P(`, ), (, ), (, `). #3. 3. between Januar 993 and March 99 and between October 99 and October 99. a) average 7, instantaneous average 3, instantaneous average 9, instantaneous average, instantaneous. positive when,,, negative when,or., and zero at,. a) t. s m/s about m/s 7. a) about 7. about.99 Both approimate the instantaneous rate of change at 3.. a) male: f () ; female: g() More females than males will have lung cancer in. The rate was changing faster for females, on average. Looking onl at 97 and, the incidence among males increased onl. per, while the incidence among females increased b 3.7. Between 99 and, the incidence among males decreased b. while the incidence among females increased b.. Since 99 is about halfwa between 99 and, an estimate for the instantaneous rate of change in 99 is the average rate of change from 99 to. The two rates of change are about the same in magnitude, but the rate for females is positive, while the rate for males is negative. Chapter Self-Test, p.. 3,,. a) positive when,and,,, negative when,, and., and zero at,, positive when,,, negative when,or,, and zero at, 3. a) Cost with card: n; Cost without card: n at least pizzas. a), # #,,or. $3. a) m. s 3 m/s. a) about (, 3) 7. Since all the eponents are even and all the coefficients are positive, all values of the function are positive and greater than or equal to for all real numbers.. a) PR # # 7,, 7 9. cm b cm b cm Chapter Getting Started, pp. 7. a) ( )( ) 3( )( ) ( 7)( 7) (3 )(3 ) e) f) (a 3)(3a ) ( 3 )(3 7 ). a) 3 s n 3 3m, m, n 3,, e) 3, 3, 3 f) a b 3b, a b, a 3b 3. a) 7,,, 3 3 3,,,, 3. a) 9, 3 3,, 3 e) f),, a, (a 3)(a )(a 3), 3,. a) 3 7. = vertical: ; horizontal: ; D PR ; R PR 7. a) translated three units to the left Answers NEL Answers

16 vertical stretch b a factor of and a horizontal translation unit to the right reflection in the -ais, vertical compression b a factor of and a, vertical translation 3 units down reflection in the -ais, vertical compression b a factor of horizontal 3, translation units right, and a vertical translation unit up. Factor the epressions in the numerator and the denominator. Simplif each epression as necessar. Multipl the first epression b the reciprocal of the second. 3(3 ) (3 ) Lesson., pp. 7. a) C; The reciprocal function is F. A; The reciprocal function is E. D; The reciprocal function is B. F; The reciprocal function is C. e) B; The reciprocal function is D. f) E; The reciprocal function is A.. a) 3 and 3 and e) no asmptotes f). and 3. a). a) = f() = f() f() 3 3 = f() = f() f() undefined 7 = f() = f() f (),. a) vertical asmptote at ; vertical asmptote at ; vertical asmptote at ; vertical asmptote at ; e) vertical asmptote at 3 ; Answers NEL

17 f) ; vertical asmptote at 3 ( 3) = f() g) vertical asmptotes 3 ; at and 7. a) = = = f() = f() h) vertical asmptotes 3 ; at and 3. a). a) D e PR f, R PR D e PR 3 f, R PR = f() = f() = 3 + = 3 + e) f) = f() = f() = f() = f() = f() Answers = f() = f() NEL Answers 3

18 9. a) D PR R PR D PR R PR #. -intercept -intercept negative on (`, ) positive on (, `) increasing on (`, `) -intercept -intercepts 3, increasing on (`,.) decreasing on (., `) negative on (`, ) and equation of reciprocal positive on (, 3) equation of = + D PR R PR -intercept 3 -intercept 3 positive on Q`, 3 R negative on Q 3, `R decreasing on (`, `) equation of reciprocal 3 = 3 = 3 D PR R PR $. -intercept -intercepts, 3 decreasing on (`,.) increasing on (., `) positive on (`, 3) and (, `) negative on (3, ) equation of reciprocal = = + = reciprocal = + 3 = +. Answers ma var. For eample, a reciprocal function creates a vertical asmptote when the denominator is equal to for a specific value of. Consider For this a b. epression, there is alwas some value of b that is that will result in a vertical a asmptote for the function. This is a graph of and the vertical asmptote 3 is at 3. Consider the function The ( 3)( ). graph of the quadratic function in the denominator crosses the -ais at 3 and and therefore will have vertical asmptotes at 3 and in the graph of the reciprocal function. However, a quadratic function, such as c, which has no real zeros, will not (3, `). have a vertical asmptote in the graph of its reciprocal function. For eample, this is the graph of a) t t If ou were to use a value of t that was less than one, the equation would tell ou that the number of bacteria was increasing as opposed to decreasing. Also, after time t, the formula indicates that there is a smaller and smaller fraction of bacteria left. e) D PR,,, R PR,, 3. a) = D PR n, R PR The vertical asmptote occurs at n. Changes in n in the f () famil cause changes in the -intercept an increase in n causes the intercept to move up the -ais and a decrease causes it to move down the -ais. Changes in n in the g() famil cause changes in the vertical asmptote of the function an increase in n causes the asmptote to move down the -ais and a decrease in n causes it to move up the -ais. n and n Answers NEL

19 . Answers ma var. For eample: ) Determine the zero(s) of the function f () these will be the asmptote(s) for the reciprocal function g(). ) Determine where the function f () is positive and where it is negative the reciprocal function g() will have the same characteristics. 3) Determine where the function f () is increasing and where it is decreasing the reciprocal function g() will have opposite characteristics.. a). 3 3 = = = Lesson., p.. a) A; The function has a zero at 3 and the reciprocal function has a vertical asmptote at 3. The function is positive for, 3 and negative for. 3. C; The function in the numerator factors to ( 3)( 3). ( 3) factors out of both the numerator and the denominator. The equation simplifies to 3, but has a hole at 3. F; The function in the denominator has a zero at 3, so there is a vertical asmptote at 3. The function is alwas positive. D; The function in the denominator has zeros at and 3. The rational function has vertical asmptotes at and 3. e) B; The function has no zeros and no vertical asmptotes or holes. f) E; The function in the denominator has a zero at 3 and the rational function has a vertical asmptote at 3. The degree of the numerator is eactl more than the degree of the denominator, so the graph has an oblique asmptote.. a) vertical asmptote at ; horizontal asmptote at vertical asmptote at 3 ; horizontal asmptote at vertical asmptote at ; horizontal asmptote at hole at 3 e) vertical asmptotes at 3and ; horizontal asmptote at f) vertical asmptote at ; horizontal asmptote at g) hole at h) vertical asmptote at ; horizontal asmptote at i) vertical asmptote at ; horizontal asmptote at j) vertical asmptote at ; hole at ; horizontal asmptote at k) vertical asmptote at 3 ; horizontal asmptote at l) vertical asmptote at ; horizontal asmptote at 3 3. Answers ma var. For eample: a) 3 e) 3 Lesson.3, pp a) A D C B. a) As S from the right, the values of f () get larger. As S from the left, the values become larger in magnitude but are negative. As S ` and as S `, f () S. e) D PR 3 R PR f) positive: (, `) negative: (`, ) g) 3. a) As S from the left, S `. As S from the right, S `. As S `, f () gets closer and closer to. e) D PR R PR 3 f) positive: (`, ) and Q, `R negative: Q, 3 R g) Answers NEL Answers

20 . a) 3; As 3, ` on the left. As 3, ` on the right. ; As, ` on the left. As, ` on the right. ; As, ` on the left. As, ` on the right. ; As, ` on the left. As, ` on the right.. a) vertical asmptote at horizontal asmptote at D PR R PR 3 -intercept f () is negative on (`, ) and positive on (, `). vertical asmptote at horizontal asmptote at D e PR f R e PR f -intercept -intercept f () is positive on (`, ) and Q and negative on Q,, `R R. Answers ma var. For eample: Answers ma var. For eample: f () 3 The function is decreasing on Q`, R and on Q The function is never, `R. increasing. hole D PR R e f Answers ma var. For eample: f () The function is decreasing on (`, ) and on (, `). The function is never increasing. vertical asmptote at horizontal asmptote at D e PR f -intercept The function will alwas be positive. = R PR -intercept f () is negative on Q`, and positive on Q R, `R a) The function is neither increasing nor decreasing; it is constant.. a) Answers ma var. For eample: f () The function is decreasing on Q`, and on Q The function R, `R. is never increasing. Answers NEL

21 The equation has a general vertical asmptote at The function has n. a general horizontal asmptote at n. The vertical asmptotes are,, and. The horizontal asmptotes, are,,, and. The function contracts as n increases. The function is alwas increasing. The function is positive on Q`, 7 and Q 3 The function n R, `R. is negative on Q 7 n, 3 R. The horizontal and vertical asmptotes both approach as the value of n increases; the - and -intercepts do not change, nor do the positive and negative characteristics or the increasing and decreasing characteristics. The vertical asmptote becomes 7 and the horizontal becomes n n.. The concentration increases over the h period and approaches approimatel.9 mg> L.. Answers ma var. For eample, the rational functions will all have vertical asmptotes at d The will all have c. horizontal asmptotes at a The will intersect the -ais at b c. The rational d. functions will have an -intercept at b a.. Answers ma var. For eample, f (). 3. f () 3 As S `, f () S `. vertical asmptote: ; oblique asmptote: 3.. Mid-Chapter Review, p. 77. a) 3 3 ; q 3 q ; z and z z ; d and 3 d 7d 3 ; 3. a) D PR; R PR; -intercept ; -intercept 3 negative on ; Q`, 3 positive on Q 3 R;, `R; increasing on (`, `) D PR; R PR.; -intercept ; -intercepts are and ; decreasing on (`, ); increasing (, `); positive on (`, ) and (, `); negative on (, ) The function is alwas increasing. The function is positive on Q`, 3 and R Q 7 The function is negative on n, `R. Q 3 The rest of the characteristics, 7 n R. do not change.. f () will have a vertical asmptote at ; g() will have a vertical asmptote at 3 f () will have a. horizontal asmptote at 3; g() will have a vertical asmptote at. 9. a) $7 $ $ No, the value of the investment at t should be the original value invested. e) The function is probabl not accurate at ver small values of t because as t S from the right, S `. f) $.. a) f () g() and h() g() D PR; R PR. ; no -intercepts; function will never be negative; decreasing on (`, ); increasing on (, `) Answers NEL Answers 7

22 D PR; R PR; -intercept ; function is alwas decreasing; positive on (`, ); negative on (, `) 3. Answers ma var. For eample: () Hole: Both the numerator and the denominator contain a common factor, resulting in for a specific value of. () Vertical asmptote: A value of causes the denominator of a rational function to be. (3) Horizontal asmptote: A horizontal asmptote is created b the ratio between the numerator and the denominator of a rational function as the function S ` and `. A continuous rational function is created when the denominator of the rational function has no zeros.. a) ; vertical asmptote hole at horizontal asmptote ; ; oblique asmptote e) and 3; vertical asmptotes. 7 7, ;, ;,. a) vertical asmptote: ; horizontal asmptote: ; no -intercept; -intercept: negative when the ; denominator is negative; positive when the numerator is positive; is negative on, ; f () is negative on (`, ) and positive on (, `); function is alwas decreasing vertical asmptote: ; horizontal asmptote: 3; -intercept: ; -intercept: f () ; function is alwas increasing; positive on (`, ) and (, `); negative on (, ) straight, horizontal line with a hole at ; alwas positive and never increases or decreases vertical asmptote: horizontal asmptote: ; -intercept: ; ; -intercept: f () ; function is alwas increasing 7. Answers ma var. For eample: Changing the function to 7 changes the graph. The function now has a vertical asmptote at and still has a horizontal asmptote at 7. However, the function is now constantl increasing instead of decreasing. The new function still has an -intercept at but now has a 7, -intercept at.. n m 3 3 ; 9. Answers ma var. For eample, f (). The graph of the function will be a horizontal line at with a hole at. Lesson., pp. 7. 3; ; Answers ma var. For eample, substituting each value for in the equation produces the same value on each side of the equation, so both are solutions.. a) 3 and 3. a) f () 3 3 f () 3 f () 3 f () 3. a) 9 3. a) 3 3 e) 9 f) 3. a) The function will have no real solutions. 3 and. and e) The original equation has no real solutions. f) and 7. a) 3.,.7.3, 7.7 e).7,.7 f).,.. a) 3 Multipl both sides of the equation b the LCD, ( )( ). ( )( ) a b ( )( ) a 3 b ( )( ) ( )( 3) Simplif. 3 Simplif the equation so that is on one side of the equation. 3 ( ) Since the product is equal to, one of the factors must be equal to. It must be because is a constant. Answers NEL

23 and 3 9. w 9.7. Machine A. min; Machine B 3. min. 7; $.. a) After.7 s The function appears to approach 9 kg>m 3 as time increases. 3. a) Tom min; Carl min; Paco min. min. Answers ma var. For eample, ou can use either algebra or graphing technolog to solve a rational equation. With algebra, solving the equation takes more time, but ou get an eact answer. With graphing technolog, ou can solve the equation quickl, but ou do not alwas get an eact answer.. 3.,.,.9,.33. a).3 and.7 (,.3) and (.7, `) Lesson., pp a) (`, ) and (3, `) (., ) and (, `). a) Solve the inequalit for. 3 # 3 # # # 3 3 # ( ) # (3, 3. a).... ( )( 3). negative:,and,, 3; positive:,,,. 3 PR,, or. 3 or (, ) or (3, `). a),,. 7,,and.3,, and.. #,and. 3 e), and 7,, f),, 7 and,. a) t,3or, t, 3 # t # or t. or t., t, 3 t, and, t, 3 e) t, and, t, f) # t,. and # t, 9. a) P(`, ) or P(, ) P(3, `) P(, ) or P(, ) P(`, 9) or P33, ) or P33, `) e) P(, ) or P(, `) f) P(`, ) or P(, `) 7. a),,.,,., Interval notation: (`, ), (.,.), (3., `) Set notation: PR,,.,,., or. 3.. a) t, and t.. It would be difficult to find a situation that could be represented b these rational epressions because ver few positive values of ield a positive value of. 9. The onl values that make the epression greater than are negative. Because the values of t have to be positive, the bacteria count in the tap water will never be greater than that of the pond water.. a) The inequalit is true for, and,,. when.. a) The first inequalit can be manipulated algebraicall to produce the second inequalit. Graph the equation 3 and determine when it is negative. The values that make the factors of the second inequalit zero are,, and. Determine the sign of each factor in the intervals corresponding to the zeros. Determine when the entire epression is negative b eamining the signs of the factors. 3. 3, ) and (, `)..,,. and,, 3.,, Lesson., pp a) ( ),. = 3+ = + - * slope a) a) slope.; vertical asmptote:. slope.7; vertical asmptote: slope.; vertical asmptote: * * * + * ( ) ( ) ( )( ) 3 slope.; vertical asmptote: Answers NEL Answers 9

24 7. a)..3. a) R().3,.3 9. a) $.7. a). 9. The number of houses that were built increases slowl at first, but rises rapidl between the third and sith months. During the last si months, the rate at which the houses were built decreases.. Answers ma var. For eample: # # ;.. a) Find s() and s(), and then solve s() s(). The average rate of change over this interval gives the object s speed. To find the instantaneous rate of change at a specific point, ou could find the slope of the line that is tangent to the function s(t) at the specific point. You could also find the average rate of change on either side of the point for smaller and smaller intervals until it stabilizes to a constant. It is generall easier to find the instantaneous rate using a graph, but the second method is more accurate. The instantaneous rate of change for a specific time, t, is the acceleration of the object at this time ;..9;. The instantaneous rate of change at (, ). The rate of change at this rate of change will be. Chapter Review, pp a) D PR; R PR; -intercept -intercept ; 3 ; alwas increasing; negative on Q`, 3 R ; positive on Q 3, `R = 3 +. a) D PR; R PR..; -intercept. and ; positive on (`, ) and (., `); negative on (,.); decreasing on (`,.); increasing on (., `) D PR; R PR. ; no -intercepts; -intercept ; decreasing on (`, ); increasing on (, `); alwas positive, never negative = + 7 = + 3. a) 7 3 horizontal asmptote; ;.; hole at ; oblique asmptote; 3 3. The locust population increased during the first.7 ears, to reach a maimum of. The population graduall decreased until the end of the ears, when the population was.. a) -intercept : horizontal asmptote: ; -intercept : vertical asmptote: ; The function is never increasing and is decreasing on (`, ) and (, `). D PR ; negative for,; positive for. D PR ; no -intercept; -intercept ; positive for ; never increasing or decreasing D PR ; no -intercept; -intercept positive for ; 3 ;.... never increasing or decreasing Answers NEL

25 .; vertical asmptote:.; D PR.; -intercept ; -intercept ; horizontal asmptote ; R PR ; positive on,. and. ; negative on.,, The function is never decreasing and is increasing on (`,.) and (., `).. Answers ma var. For eample, consider the function f () You know that. the vertical asmptote would be. If ou were to find the value of the function ver close to ( sa f (.99) or f (.)) ou would be able to determine the behaviour of the function on either side of the asmptote. f (.99) (.99) f (.) (.) To the left of the vertical asmptote, the function moves toward `. To the right of the vertical asmptote, the function moves toward `. 7. a). and 3 or and 3. about min 9.. das and 3.97 das. a),3and.73,,.73,, and,,,.33 and,,,,...7, t, and t..73. a) ; 3.; and 3. a). mg> L> h. mg> L> h The concentration of the drug in the blood stream appears to be increasing most rapidl during the first hour and a half; the graph is steep and increasing during this time.. and ; a) As the -coordinate approaches the vertical asmptote of a rational function, the line tangent to graph will get closer and closer to being a vertical line. This means that the slope of the line tangent to the graph will get larger and larger, approaching positive or negative infinit depending on the function, as gets closer to the vertical asmptote. As the -coordinate grows larger and larger in either direction, the line tangent to the graph will get closer and closer to being a horizontal line. This means that the slope of the line tangent to the graph will alwas approach zero as gets larger and larger. Chapter Self-Test, p. 3. a) B A. a) If f (n) is ver large, then that would make a ver small fraction. f (n) If f (n) is ver small (less than ), then that would make ver large. f (n) If f (n), then that would make undefined at that point because f (n) ou cannot divide b. If f (n) is positive, then that would make also positive because ou are f (n) dividing two positive numbers kg; $./kg. a) Algebraic; and 3 Algebraic with factor table The inequalit is true on (,.) and on (,.).. a) To find the vertical asmptotes of the function, find the zeros of the epression in the denominator. To find the equation of the horizontal asmptotes, divide the first two terms of the epressions in the numerator and denominator. This tpe of function will have a hole when both the numerator and the denominator share the same factor ( a). Chapter Getting Started, p. 3. a) 33. a) P(3, ) sin u cos u3 tan u,, 3, csc u sec u cot u 3, 3, a)!3!3 e)! f ). a), 3 3,, e) 3, 3 f) 9. a) period 3 ; amplitude ; ; R PR # # period 3 ; amplitude ; ; R PR # #. a) period ; ; o to the left; amplitude Answers NEL Answers