M47_Chapter_2-2.pdf - Nguyen Kemp Long Beach City College...

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Chapter 2.2 First Derivative. Critical Points. Relative extreme values Nguyen, Kemp - Long Beach City College - Copyright © 2020 2 2.2 - FIRST DERIVATIVE. CRITICAL POINTS. RELATIVE EXTREME VALUES RATIONAL FUNCTIONS Rational Functions and Curve Sketching In this section we will graph Rational Functions. Rational Functions unlike polynomial functions are the ratio of two independent functions. The rational function is a two-fold function in that it has an independent numerator function ℎ(?) and an independent denominator function ?(?) . ? = ?(?) = ℎ(?) ?(?) = ? − 3 ? + 5 When we analyze rational functions, we find or identify the function’s Vertical Asymptote and its Horizontal Asymptote. If we have a graph of the Rational Function, ( ? ), we will see what we can conclude about the values of ( ? ). Likewise, if we know values of ( ? ), we will see what we can conclude about the graph of ( ? ). ?(?) = ? − 3 ? + 5
Chapter 2.2 First Derivative. Critical Points. Relative extreme values Nguyen, Kemp - Long Beach City College - Copyright © 2020 3 Definitions: A rational function is a function of the form ? = ?(?) = ℎ(?) ?(?) where both h(x) and g(x) are independent functions. Graphs of rational functions have vertical and sometimes horizontal asymptotes. Test for Vertical Asymptote , at ( ? = 𝑎 ), such that [ ?(𝑎) = 0 𝑎?? ℎ(𝑎) ≠ 0 ] ? = ?(𝑎) = ℎ(𝑎) 0 = ?????𝑖??? 𝑽𝒆??𝒊𝒄𝒂? 𝑨???????𝒆 → ?𝐴: ? = 𝑎 The Rational Function ( ? ) has a Vertical Asymptote at any value ( ? = 𝑎 ), when [ ?(𝑎) = 0 ] and [ ℎ(𝑎) ≠ 0 ]. If both ?(𝑎) = 0 and ℎ(𝑎) = 0 , then the function has a hole in the graph at ? = 𝑎. The Horizontal Asymptote exits when the Rational Function ( ? ) is: ? = lim 𝑥 → ∞ ?(?) = ℎ(∞) ?(∞) = ? and, or ? = lim 𝑥 → −∞ ?(?) = ℎ(−∞) ?(−∞) = ? 𝑯??𝒊????𝒂? 𝑨???????𝒆 → ?𝐴: ? = ?
Chapter 2.2 First Derivative. Critical Points. Relative extreme values Nguyen, Kemp - Long Beach City College - Copyright © 2020 4 The Rational Function ( ? ) has a Horizontal Asymptote if the limit as ( ? ) approaches infinity is a value ( ? = ? ). Since we have not discussed much finding limits at infinity, such as lim 𝑥 → ∞ ?(?) , recall from your College Algebra classes a different way of determining Horizontal asymptotes. Horizontal Asymptote of Rational Functions The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator. Degree of denominator > degree of numerator: Horizontal asymptote at 0 y = Degree of denominator = degree of numerator: Horizontal asymptote at ratio of leading coefficients. Degree of denominator < degree of numerator: No horizontal asymptote
Chapter 2.2 First Derivative. Critical Points. Relative extreme values Nguyen, Kemp - Long Beach City College - Copyright © 2020 5 Critical Numbers Rational Functions
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