Rational FunctionsDefinitionLong Run BehaviorExampleSlide 5Try This OneWhen Numerator Has Larger DegreeSummarizeExtra InformationTry It OutApplicationAssignmentRational FunctionsLesson 9.4Definition•Consider a function which is the quotient of two polynomials•Example: ( )( )( )P xR xQ x=Both polynomials2500 2( )xr xx+=Long Run Behavior•Given•The long run (end) behavior is determined by the quotient of the leading termsLeading term dominates forlarge values of x for polynomialLeading terms dominate forthe quotient for extreme x11 1 011 1 0...( )...n nn nm mm ma x a x a x aR xb x b x b x b----+ + + +=+ + + +nnmma xb xExample•Given•Graph on calculatorSet window for -100 < x < 100, -5 < y < 5223 8( )5 2 1x xr xx x+=- +Example•Note the value for a large x•How does this relate to the leading terms?2235xxTry This One•Consider•Which terms dominate as x gets large•What happens to as x gets large?•Note:Degree of denominator > degree numeratorPrevious example they were equal25( )2 6xr xx=+252xxWhen Numerator Has Larger Degree•Try•As x gets large, r(x) also gets large•But it is asymptotic to the line22 6( )5xr xx+=25y x=SummarizeGiven a rational function with leading terms•When m = nHorizontal asymptote at•When m > nHorizontal asymptote at 0•When n – m = 1Diagonal asymptote nnmma xb xabay xb=Extra Information•When n – m = 2Function is asymptotic to a parabola•The parabola isWhy?nnmma xb x3 22 5 65x x xyx- - +=22 115 5y x x= - -Try It Out•ConsiderWhat long range behavior do you predict?What happens for large x (negative, positive)What happens for numbers close to -4?2( )4xG xx=+x -100 -10 10 50 100 1000G(x)x -4.2 -4.1 -4.01 -3.99 -3.9 -3.8G(x)Application•Cost to manufacture n units isC(n) = 5000 + 50n•Average cost per unit is•What is C(1)? C(1000)?•What is A(1)? A(1000)?•What is the trend for A(n) when n gets large?( )( )C nA nn=Assignment•Lesson 9.4•Page 413•Exercises 1 – 21
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