Understanding Function Analysis in Calculus

mastermathmentor.com presents CALCULUS ON THE WALL Helping students learn and teachers teach 6. Function Analysis (Sample) Created by: Stu Schwartz Un-narrated Version Edited by: Jake Weinberg Graphics: Apple Grapher: Version 2.3 Math Type: Version 6.7 Intaglio: 2.9.5a Fathom: Version 2.11

Relative Extrema Critical Points are defined as x-values where f (x) = 0 or f (x) does not exist. If f (x) changes from negative to positive at c, then there is a relative (local) minimum at x = c, and f(c) is a relative minimum of f. If f (x) changes from positive to negative at c, then there is a relative (local) maximum at x = c, and f(c) is a relative maximum of f. c c f has a relative minimum at x = c f has a relative maximum at x = c Finding relative extrema by the signs of f (x) is called the first derivative test. www.mastermathmentor.com

Example Find the intervals in which the function is concave up and concave down, any inflection points, and graph the function. a. f(x) = 1 + 6x2 2x3 x ( )=12x-6x2= 0 6x 2- x ( f x ( ) has critical points at x = 0,x = 2 sign f chart 0 2 f )= 0 x ( ):----0+++++0----- f is increasing on (0, 2) f is decreasing on (- ,0) and (2, ) There is a relative minimum at (0, 1) There is a relative maximum at (2, 9) x ( ) =12-12x = 0 ) = 0 x ( ) has critical points at x =1 x ( ):++++++------- f 12 1- x ( f sign f chart 1 f is concave up on (- , 1) and concave down on (1, ). There is an inflection pt. at x = 1. www.mastermathmentor.com

Going Backwards Typically, given a function you can find its first and second derivatives. But, if we are given a graph of f (x), the derivative of a function f(x), we can sketch a graph of a possible function fitting that information. Given f (x), sketch a possible f(x) Since f (x) is the slope of the given f (x), and the slope is positive, we make a sign chart of f (x): x ( ):+++++++++++++ f is concave up on - , f ( ) Since we have f (x), we find that there is a critical point at x = 1. x ( ):------0++++++ 1 f is decreasing on - ,1 ( f has a relative minimum at x =1 f ) and increasing on 1, ( ) Any of these graphs fit the criteria. www.mastermathmentor.com

Function Analysis & Related Rates Functions can represent changing information over time f(t) and thus its derivative f (t) and 2nd derivative f (t) have meanings in context. p t ( )= 0: 4,5 ( ) 1. Price is steady. Let p(t) represents the price of gasoline on day t of a specific week. This graph is y = p (t). Determine the time or interval of time: p t ( )>0: 0,4 ( ) 2. Price is increasing. 3. Rate of change of price is positive and constant. p t ( )= +C: 2,3 ( ) 4. Rate of change of price is increasing. p ( ) t ( )= 0,2 ( p t ( )> 0: ) ), 6,7 ( p t ( ) switches from positive to negative: 4,5 5. Price is at a maximum. ( ) p t ( ) has its smallest value: day 6 6. Change of price is at a minimum. www.mastermathmentor.com

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