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Unformatted text preview: 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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