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4.3 Derivatives - Shape of Graph

Course: M 241, Fall 2009
School: Delaware
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Word Count: 255

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241 Section MATH 4.3 Page 1 Increasing/Decreasing Test a. If f (x) &gt; 0 on an interval, then f is increasing on the interval. b. If f (x) &lt; 0 on an interval, then f is decreasing on the interval. The First Derivative Test Suppose that c is a critical number of a continuous function f. 1. If f changes from +'ve to 've at c, then f has a local max at c. 2. If f changes from 've to +'ve at c, then...

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241 Section MATH 4.3 Page 1 Increasing/Decreasing Test a. If f (x) > 0 on an interval, then f is increasing on the interval. b. If f (x) < 0 on an interval, then f is decreasing on the interval. The First Derivative Test Suppose that c is a critical number of a continuous function f. 1. If f changes from +'ve to 've at c, then f has a local max at c. 2. If f changes from 've to +'ve at c, then f has a local min at c. 3. If f does not change sign at c, then f has no local max or min at c. Definition Concave Upward: The graph of f lies above all its tangents on an interval Concave Downward: The graph of f lies below all its tangents an on interval MATH 241 Section 4.3 Page 2 Concavity Test a. If f (x) > 0 for all x in an interval, then the graph of f is concave upward on the interval b. If f (x) < 0 for all x in an interval, then the graph of f is concave downward on the interval Definition A point P on a curve y = f (x) is called an inflection point if f is continuous at P and the curve changes c...

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