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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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Definition of Curve Sketching Curve Sketching is the process of using the first and second derivative and information gathered from the original equation.

Definition of Curve Sketching Curve Sketching is the process of using the first and second derivative and information gathered from the original equation.

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