1. 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]

2. 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)

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)

4. 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]

5. 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

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

7. 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

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

9. 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

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