4_3 - Section 4.3 How Derivatives Aect the Shape of a Graph...

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Section 4.3: How Derivatives Affect the Shape of a Graph Instructor: Ms. Hoa Nguyen ([email protected]) Because f ( x ) represents the slope of the curve y = f ( x ) at the point ( x, f ( x )), it tells us the direction in which the curve proceeds at each point. Increasing/ Decreasing Test 1. f ( x ) > 0 on an interval f is increasing on that interval. 2. f ( x ) < 0 on an interval f is decreasing on that interval. Example 1 of Section 4.3 (textbook): The First Derivative Test Suppose c is a critical number of a continuous function f 1. f changes from positive to negative at c f has a local maximum at c . 2. f changes from negative to positive at c f has a local minimum at c . 3. f does not change sign at c f has NO local maximum or minimum at c . Example 2 of Section 4.3 (textbook): Example 3 of Section 4.3 (textbook): Concavity: Concave Upward/ Downward If the graph of f lies above all of its tangents on an interval I , then it is called concave upward on I .
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