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Graphing Using Derivatives: Increasing/Decreasing Theorem

Learn how to graph functions using derivatives and the increasing/decreasing theorem. Use the first and second derivative tests to find local extrema and determine concavity and points of inflection.

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Graphing Using Derivatives: Increasing/Decreasing Theorem

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  1. Math 1304 4.3 – Graphing Using Derivatives

  2. Increasing/Decreasing Theorem: (a) If f’(x) > 0 on an interval, then f is increasing on that interval. (b) If f’(x) < 0 on an interval, then f is decreasing on that interval. Proof: Use MVT. • Let x1 < x2 be different numbers in the interval. Then there is a c in (x1, x2) such that f’(c) = (f(x2)-f(x1))/(x2-x1). But f’(c)>0. So (f(x2)-f(x1))/(x2-x1) > 0. So f(x2)-f(x1) > 0. Thus f(x2) > f(x1). (b) Left to students.

  3. First Derivative Test • Theorem (First Derivative Test) Suppose that c is a critical point 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, it has neither maximum nor mininimum at c.

  4. Concavity • Definition of concave upward: If the graph of a function f is above all its tangents on an interval I, then it is called concave upward on I. • Definition of concave downward: If the graph of a function f is below all its tangents on an interval I, then it is called concave downward I.

  5. Inflection Point • Definition: A point P on a curve y = f(x) is called an inflection point if f is continous there and the curve changes concavity there.

  6. Second Derivative Test • Theorem (second derivative test) If f is function for which f” is continuous in an interval around a point c, then • If f’(c) = 0 and f”(c) > 0, then f has a local minimum at c. • If f’(c) = 0 and f”(c) < 0, then f has a local maximum at c.

  7. Graphing Method • Start with a function f(x). • Compute formulas for f’(x) and f”(x). • Find all points of discontinuity, points where the first derivative is zero or undefined (critical points), and all points where the second derivative is zero (possible points of inflection) • Make a table of f(x), f’(x), and f”(x) for all these interesting points, making room for test points in between interesting points. • Use the first derivative to find increasing/decreasing behavior • Use the second derivative to determine concavity and points of inflection. • Use the first and second derivative tests to find all local extrema. • Plot this information and sketch the graph.

  8. Examples • See book and HW

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