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3.1: Increasing and Decreasing Functions

3.1: Increasing and Decreasing Functions. Definition. A function f is increasing on an interval if for any 2 numbers x 1 and x 2 in the interval x 1 <x 2 implies f(x 1 ) < f(x 2 )

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3.1: Increasing and Decreasing Functions

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  1. 3.1: Increasing and Decreasing Functions

  2. Definition • A function f is increasing on an interval if for any 2 numbers x1 and x2 in the interval x1<x2 implies f(x1) < f(x2) • A function f is decreasing on an interval if for any 2 numbers x1 and x2 in the interval x1<x2 implies f(x1) > f(x2)

  3. Look at y = x2. What do you notice about the slopes?

  4. Increasing/Decreasing Test • If f’(x) > 0 for all x in the interval (a, b), then f is increasing on the interval (a, b). • If f’(x) < 0 for all x in the interval (a, b), then f is decreasing on the interval (a, b). • If f’(x) = 0 for all x in the interval (a, b), then f is constant on the interval (a, b).

  5. Use the I/D Test for y = x2. What is the derivative? Where is the derivative positive? Where is the derivative negative?

  6. Checkpoint 2 p. 185

  7. Critical Numbers • If f is defined at c, then c is a critical number of f if f’(c) = 0 or f’(c) is undefined.

  8. To Apply the I/D Test • Find f’(x) • Locate critical numbers • Set up a number line, test x-values in each interval

  9. Example • Find the intervals on which f(x) =x3 – 12x is increasing and decreasing.

  10. Example • Find the intervals on which is increasing and decreasing.

  11. Now you try: • Determine the intervals on which the following functions are increasing/decreasing.

  12. Last Example • Checkpoint 6 p. 190

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