Lect25 - Derivatives and the Shapes of Graphs 1 1 f ( x) =...

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92.131 Lecture 25 1 of 22 Ronald Brent © 2009 All rights reserved. x - 5- 4- 3- 2- 1 12345 -5 -4 -3 -2 -1 1 2 3 y 4 5 x - 5 - 4 - 3 - 2 - 1 -5 -4 -3 -2 -1 1 y 2 3 4 5 x -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 y 4 5 Derivatives and the Shapes of Graphs x x x x f 2 2 1 3 1 ) ( 2 3 + = 2 ) ( 2 + = x x x f 1 2 ) ( + = x x f B A C C A B B ’’
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92.131 Lecture 25 2 of 22 Ronald Brent © 2009 All rights reserved. What ) ( x f says about ) ( x f . The sign of ) ( x f . Determines where the function ) ( x f increases or decreases. Notice that the function rises to the left of A , falls between A and C , then rises again to the right of C . This information is given in the graph of ) ( x f . Notice that the function ) ( x f is positive to the left of A ’, and the right of C . Between A and C the function ) ( x f is negative. Stationary, Maximum, and Minimum Points: The points A and C where the function ) ( x f is relatively high and low occur where 0 ) ( = x f , A and C . These are called stationary points. Concavity and Inflection: At the point B , the function’s concavity changes from concave down to concave up. This corresponds to the point where the graph of ) ( x f changes from decreasing to increasing.
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92.131 Lecture 25 3 of 22 Ronald Brent © 2009 All rights reserved. Increasing and Decreasing Test 1. If 0 ) ( > x f on an interval, then ) ( x f increases on that interval. 2. If 0 ) ( < x f on an interval, then ) ( x f decreases on that interval. First Derivative Test Suppose that c is a critical number of a continuous function f . If 0 ) ( > x f for c x < , and 0 ) ( < x f for c x > , then c is a local maximum point. i.e. If f changes from positive to negative at c . If 0 ) ( < x f for c x < , and 0 ) ( > x f for c x > , then c is a local minimum point. i.e. If f changes from negative to positive at c . If f does not change sign at c , then f has no local extreme value at c .
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92.131 Lecture 25 4 of 22 Ronald Brent © 2009 All rights reserved. Concavity, and Inflection Points The graph of ) ( x f is concave up at c x = if the slope function ) ( x f is increasing at c x = . The graph of ) ( x f is concave down at c x = if the slope function ) ( x f is decreasing at c x = . A point where the function changes concavity is called an inflection point . Concavity Test If 0 ) ( > x f for all x in I , then the graph of f is concave upward on I .
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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Lect25 - Derivatives and the Shapes of Graphs 1 1 f ( x) =...

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