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Page 44 What does the 1 st derivative of a function tell us about the graph of the function? It tells us __________________________________________________________ What does the 2 nd derivative of a function tell us about the graph of the function? It tells us __________________________________________________________ Therefore, to sketch the graph of a function, f(x), we should, For where the function f(x) is increasing/decreasing and attains its local extrema: 1.Find the 1 st derivative, f (x). 2.Find the x-values such that f (x) = 0, and sometimes, also find the x-values such that f (x) doesnt exist. * 3.Determine the intervals for which f is increasing and decreasing, determine the locations of its local extrema, if any. For where the function f(x) is concave up/concave down and attains its inflection points: 1. Find the 2 nd derivative, f (x). 2. Find the x-values such that f (x) = 0, and sometimes, also find the x-values such that f (x) doesnt exist. 3. Determine the intervals for which f is _______________________________________________________________ Example: f(x) = 1 / 3 x 3 + x 2 – 8x + 5 Sketching the Graph of a Function Using Its 1st and 2nd Derivatives Intervals(, ) x f (x) Inc/Dec? Graph Intervals(, ) x f (x) CU/CD? Graph *Definition: A critical number of a function f is a number c in the domain of f such that either f (c) = 0 or f (c) doesnt exist.

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Page 45 If a function is not a constant function, then it will increase and/or _____________. If it is not a linear function, it will be concave up and/or __________. If so, the graph of the function can only consist of one or more of the following 4 pieces: Example: f(x) = x 4 – x 2 – 2x – 1 (Note: The only critical number from f (x) is x = 1.) Inc and CUDec and ________ and ____Dec and CD Sketching the Graph of a Function (contd) Intervals( ) x f (x) Inc/Dec? f (x) CU/CD? Graph f at Key Numbers Max/Min/Inf

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Page 46 When sketching the graph of a function, f(x), besides considering increasing/decreasing and concavity (i.e., concave up/concave down), we also need to considering following: A. Domain: determine all possible values of x B. Intercepts: y-intercept (by plug __ into f(x)) and x-intercept(s) (by setting f(x) = __ and solve for x) * C. Symmetry: determine whether it is symmetric with respect to (wrt) the ______ or wrt the ______ (see below.) D. Asymptotes: determine whether there is any _______ and/or ______ asymptotes (see below.) * Only when the x-intercepts are manageable to find. Of course, for where f(x) is increasing/decreasing (incl. local extrema) and concavity (incl. inflection points), we have to find the following: E. Intervals of Increase/Decrease: Use the I/D Test: f (x) > 0 increasing and _________ __________ F. Local Maximum/Minimum: Find the x-values where f (x) = 0 or f (x) doesnt exist. f will likely have local extrema at these x-values (but not a must). G. Concavity and Inflection Points: Use the Concavity Test: f (x) > 0 concave up and ______ _____. Find the x-values where f (x) = 0 or f (x) doesnt exist. f will likely have inflection points at these x-values (but not a must). When you have all these components, H. Sketch the function. General Techniques/Considerations When Sketching a Function C. SymmetryD. Asymptotes

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Page 47 Ex 1. f(x) = (x 2 – 4)/(x 2 + 1) A. Domain: B. Intercepts: C. Symmetry: D. Asymptotes: E. Intervals of Increase/Decrease: F. Local Maximum/Minimum: G. Concavity and Inflection Points: H. Sketch the function Sketch a Function Using A-H (from the Previous Page) Intervals( ) x f (x) Inc/Dec? f (x) CU/CD? Graph f at Key Numbers Max/Min/Inf/VA

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Page 48 Ex 1. f(x) = x/(x 2 – 4) A. Domain: B. Intercepts: C. Symmetry: D. Asymptotes: E. Intervals of Increase/Decrease: F. Local Maximum/Minimum: G. Concavity and Inflection Points: H. Sketch the function Sketch a Function Using A-H Intervals( ) x f (x) Inc/Dec? f (x) CU/CD? Graph f at Key Numbers Max/Min/Inf/VA

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