End Behavior Models Section 2.2b.

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Presentation transcript:

End Behavior Models Section 2.2b

End Behavior Models For large values of x, we can sometimes model the behavior of a complicated function by a simpler one that acts in virtually the same way… Ex: Given: Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x| large. First, graph in by What do you notice? Next, graph in by How about now?

End Behavior Models Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x| large. We can test our claim algebraically by examining the ratio of the two functions as x approaches positive or negative infinity: Convincing Evidence???

Definition: End Behavior Model The function g is (a) a right end behavior model for f if and only if (b) a left end behavior model for f if and only if Note: A function’s right and left end behavior models need not be the same function!!! (but when a function fits both, it is simply called an end behavior model)

Guided Practice For all polynomials, an end behavior model is simply the leading term! For all rational functions, an end behavior model is simply the ratio of the leading terms from the numerator and denominator! Then simply use some logical thinking to establish the existence of any horizontal asymptotes…

Guided Practice E.B.M.: H.A.: None For each of the following, (a) find a power function end behavior model for f, and (b) identify any horizontal asymptotes. E.B.M.: H.A.: None

Guided Practice E.B.M.: H.A.: For each of the following, (a) find a power function end behavior model for f, and (b) identify any horizontal asymptotes. E.B.M.: H.A.:

Guided Practice E.B.M.: H.A.: None For each of the following, (a) find a power function end behavior model for f, and (b) identify any horizontal asymptotes. E.B.M.: H.A.: None

Guided Practice E.B.M.: H.A.: For each of the following, (a) find a power function end behavior model for f, and (b) identify any horizontal asymptotes. E.B.M.: H.A.:

Guided Practice Find the limit of f (x) as x approaches negative and positive infinity, and as x approaches zero from the left and from the right.