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2311lectureoutline4.3

# 2311lectureoutline4.3 - I it is called concave downward on...

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MAC2311, Calculus I 4.3 How Derivatives Affect the Shape of a Graph Increasing/Decreasing Test: If f’(x) > 0 on an interval, then f is increasing on that interval. If f’(x) < 0 on an interval, then f is decreasing on that interval. The First Derivative Test: Suppose that c is a critical number of a continuous function f. If f’ changes from positive to negative at c , then f has a local maximum at c . If f’ changes from negative to positive at c , then f has a local minimum at c . If f’ does not change sign at c , then f has a no local maximum or minimum at c. Definition: If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I . If the graph of f lies below all of its tangents on
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Unformatted text preview: I , it is called concave downward on I . Concavity Test: If f(x) &amp;gt; 0 for all x in I , then the graph of f is concave upward on I . If f(x) &amp;lt; 0 for all x in I , then the graph of f is concave downward on I. Definition: A point P on a curve y = f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P . The Second Derivative Test: Suppose f is continuous near c . If f(c) = 0 and f(x) &amp;gt; 0, then f has a local minimum at c. If f(c) = 0 and f(x) &amp;lt; 0, then f has a local maximum at c. Try exercises 2, 6, 10, 20, 36, 46, 50 on pp. 295-296 in your text....
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