MATH 140 Lecture 24: Graphing Problems
6 Nov 2015
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Horizontal asymptotes: the line y = b is a horizontal asymptote of the function f if f (x)=b lim x or lim x f (x)=b or both, note, a horizontal asymptote is a line of the form y = b, a function can have 0, 1, or 2 horizontal asymptotes, there can be infinitely many vertical asymptotes. Horizontal: lim x f (x)= lim x f (x )=1 y=1 is a horizontal asymptote: graph: 2. g(x)= x2 (2 x)2 g"( x)= 4x (2 x)3 g" "( x)= 8x +8 (2 x)4: asymptotes: Horizontal: y = 1 g" (0)=0 g" "(0)>0 g (0)=relative minimum o o: the second derivative is negative when x < 1, positive on 12. The graph of g is therefore concave downward on concave upward on ( 1,2) (2, : graph: ( , 1) and, note: a graph may cross its horizontal asymptote(s) infinitely many times.
