Increasing, Decreasing, Constant

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Presentation transcript:

Increasing, Decreasing, Constant Objectives: Be able to determine where a function is increasing, decreasing or constant with the use of calculus. Critical Vocabulary: Increasing, Decreasing, Constant Warm Up: Use the graphs to determine where the graph is increasing, decreasing, or constant. [answers should be in interval notation]

A function f is increasing on an interval for any 2 numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2). Since x1 < x2 (1 < 2) that implies f(x1) < f(x2) (0 < 3)

A function f is increasing on an interval 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 for any 2 numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2). Since x1 < x2 (-6 < -4) that implies f(x1) > f(x2) (3 > -3)

How to test for Increasing and Decreasing using CALCULUS Locate the ___________________ to determine our boundaries for our test intervals. 2. Determine the sign of _______ at 1 test value in each interval. 3. Use the following theorem to determine if that interval is _____________ or _________________. If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b): 1. If f’(x) > 0 for all x in (a, b), then f is ________ on [a,b] 2. If f’(x) < 0 for all x in (a, b), then f is ________ on [a,b] 3. If f’(x) = 0 for all x in (a, b), then f is ________ on [a,b]

Example 1: Find the open intervals on which the following function is increasing or decreasing. 1st: Find the critical numbers Increasing: ___________ Decreasing: ___________ Constant: ____________ 2nd: Test Values Interval Test Value Sign of f’(x) Conclusion

Example 1: Find the open intervals on which the following function is increasing or decreasing. Increasing: ___________ Decreasing: ___________ Constant: ____________

Example 2: Find the open intervals on which the following function is increasing or decreasing. 1st: Find the critical numbers Increasing: ___________ Decreasing: ___________ Constant: ____________ 2nd: Test Values Interval Test Value Sign of f’(x) Conclusion

Example 2: Find the open intervals on which the following function is increasing or decreasing. Increasing: ___________ Decreasing: ___________ Constant: ____________

A function is STRICTLY MONOTONIC on an interval if it is either increasing or decreasing on the entire interval. Example 3: Find the open intervals on which the function f(x) = x3 increasing or decreasing. 1st: Find the critical numbers Increasing: ___________ Decreasing: ___________ Constant: ____________ 2nd: Test Values Interval Test Value Sign of f’(x) Conclusion

Page 334 #1-9 odds (MUST USE CALCULUS!!!!)