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curve_sketching_from_online_tutorial_new

curve_sketching_from_online_tutorial_new - The First...

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The First Derivative: Maxima and Minima Refer to: http://www.math.hmc.edu/calculus/tutorials/extrema Harvey Mudd College Online Tutorial for Calculus Consider the function f ( x ) = 3 x 4 4 x 3 12 x 2 + 3. It is difficult to find the regions of which f is increasing or decreasing, relative maxima or minima, or the absolute maximum or minimum value of f by inspection. Graphing by hand is tedious and imprecise. Even the use of a graphing program will only give us an approximation for the locations and values of maxima and minima. We can use the first derivative of f , however, to find all these things quickly and easily. First Derivative Test - Increasing or Decreasing? (See Bus 14B textbook p. 731) If a function has a positive first derivative, the function is increasing. (As x increases in value, y is getting larger. This is like going UP on a roller coaster.) If a function has a negative first derivative, the function is decreasing. (As x increases in value, y is getting smaller. This is like going DOWN on a roller coaster.) Example 1: Given: f ( x ) = 3 x 4 4 x 3 12 x 2 + 3 Function is continuous, no restrictions on the domain. First Derivative Test – Increasing/Decreasing 1) Take the first derivative. f ’( x ) = 12 x 3 12 x 2 24 x (no values where f’ is undefined) 2) Obtain the Critical X-Values a) Set the first derivative equal to zero.
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