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Sketching the Graph of a Function Using Its 1st and 2nd Derivatives Page 44 What does the 1st derivative of a function tell us about the graph of the function? It tells us __________________________________________________________ What does the 2nd 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 1st derivative, f (x). 2. Find the x-values such that f (x) = 0, and sometimes, also find the x-values such that f (x) doesn’t 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 2nd derivative, f (x). 2. Find the x-values such that f (x) = 0, and sometimes, also find the x-values such that f (x) doesn’t exist. 3. Determine the intervals for which f is _______________________________________________________________ Example: f(x) = 1/3 x3 + x2 – 8x + 5 *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) doesn’t exist. 20 Intervals ( , ) ( , ) ( , ) Intervals x x f (x) f (x) Inc/Dec? CU/CD? Graph Graph ( , ) ( , ) ( , ) 15 10 5 -4 -2 2 -5 -10 -15 -20 4 6 Sketching the Graph of a Function (cont’d) 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: Inc and CU Dec and ____ ____ and ____ Dec and CD Example: f(x) = x4 – x2 – 2x – 1 (Note: The only critical number from f (x) is x = 1.) Intervals ( x f (x) Inc/Dec? 4 f (x) 3 2 CU/CD? 1 Graph -3 -2 -1 1 2 3 -1 -2 f at Key Numbers -3 -4 Max/Min/Inf ) ( ) ( ) ( ) ( ) General Techniques/Considerations When Sketching a Function 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) doesn’t 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) doesn’t 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. C. Symmetry D. Asymptotes Sketch a Function Using A-H (from the Previous Page) Page 47 Ex 1. f(x) = (x2 – 4)/(x2 + 1) A. Domain: B. Intercepts: C. Symmetry: D. Asymptotes: E. Intervals of Increase/Decrease: Intervals F. Local Maximum/Minimum: ( x G. Concavity and Inflection Points: f (x) H. Sketch the function Inc/Dec? f (x) 5 4 CU/CD? 3 Graph 2 1 f at Key Numbers -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 2 3 4 5 Max/Min/Inf/VA ) ( ) ( ) ( ) ( ) Sketch a Function Using A-H Page 48 Ex 1. f(x) = x/(x2 – 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 Intervals 5 x 4 f (x) ( 3 Inc/Dec? 2 f (x) 1 -5 -4 -3 -2 -1 1 -1 2 3 4 5 CU/CD? Graph -2 -3 -4 -5 f at Key Numbers Max/Min/Inf/VA ) ( ) ( ) ( ) ( )