slides12 - MAT1300 Lecture 11 Concavity and Related Rates...

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Unformatted text preview: MAT1300 Lecture 11 Concavity and Related Rates Pieter Hofstra October 27, 2009 Overview Second Derivative Related Rates 1 Second Derivative Concavity Second Derivative Test 2 Related Rates Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test Intuition: a curve is concave up on an interval I if it looks like on I . It is concave down on I if it looks like . We need a more precise definition. Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test Intuition: a curve is concave up on an interval I if it looks like on I . It is concave down on I if it looks like . We need a more precise definition. So f is increasing on this interval, and f has a local minimum. Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test Here f is decreasing on this interval and f has a local maximum. Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test This suggests the following definition of concavity: Definition If f is a function, we say f is concave up on I if f is increasing on I . We say that f is concave down on I if f is decreasing on I . Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test This suggests the following definition of concavity: Definition If f is a function, we say f is concave up on I if f is increasing on I . We say that f is concave down on I if f is decreasing on I . Since we can determine whether a function is increasing or decreasing by looking at its derivative, we consider the derivative of f ( x ), denoted f 00 ( x ), the second derivative of f . Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test This suggests the following definition of concavity: Definition If f is a function, we say f is concave up on I if f is increasing on I . We say that f is concave down on I if f is decreasing on I . Since we can determine whether a function is increasing or decreasing by looking at its derivative, we consider the derivative of f ( x ), denoted f 00 ( x ), the second derivative of f . We have: Theorem Let f be a function. Then 1 If f 00 ( x ) > for all x in I, then f is concave up on I. 2 If f 00 ( x ) < for all x in I, then f is concave down on I. Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test The characterization of concavity in terms of the second derivative leads to the following result: Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity Second Derivative Test The characterization of concavity in terms of the second derivative leads to the following result: Theorem (The Second-Derivative Test): Suppose that c is a critical point for the function f . Then Pieter Hofstra MAT1300 Lecture 11 Overview Second Derivative Related Rates Concavity...
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This note was uploaded on 01/07/2010 for the course MAT mat1300 taught by Professor Pieterhofstra during the Fall '09 term at University of Ottawa.

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slides12 - MAT1300 Lecture 11 Concavity and Related Rates...

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