Increasing and decreasing functions ap calc sec 3.3
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The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
Increasing and decreasing functions ap calc sec 3.3
- 1. Increasing and Decreasing Functions and the First Derivative Test AP Calculus – Section 3.3 Objectives: 1.Find the intervals on which a function is increasing or decreasing. 2.Use the First Derivative Test to classify extrema as either a maximum or a minimum.
- 2. Increasing and Decreasing Functions • The derivative is related to the slope of a function
- 3. Increasing and Decreasing Functions On an interval in which a function f is continuous and differentiable, a function is… increasing if f ‘(x) is positive on that interval, ( f ‘ (x) > 0 ) decreasing if f ‘(x) is negative on that interval, and ( f ‘ (x) < 0 ) constant if f ‘(x) = 0 on that interval.
- 4. Visual Example f ‘(x) < 0 on (-5,-2) f(x) is decreasing on (-5,-2) f ‘(x) = 0 on (-2,1) f(x) is constant on (-2,1) f ‘(x) > 0 on (1,3) f(x) is increasing on (1,3)
- 5. Finding Increasing/Decreasing Intervals for a Function To find the intervals on which a function is increasing/decreasing: 1.Find critical numbers. - These determine the boundaries of your intervals. 2.Pick a random x-value in each interval. 3.Determine the sign of the derivative on that interval.
- 6. Example Find the intervals on which the function 3 f ( x) = x − x is increasing and decreasing. 2 3 2 Critical numbers: f ' ( x) = 3x 2 − 3 x 3x 2 − 3x = 0 3 x( x − 1) = 0 x = {0,1}
- 7. Example Test an x-value in each interval. Interval Test Value f ‘(x) (−∞,0) (0,1) (1, ∞) −1 1 2 2 f ' (−1) = 6 3 1 f ' = − 4 2 f ' ( 2) = 6 f(x) is increasing on (−∞,0) and (1, ∞) . f(x) is decreasing on (0,1).
- 8. Practice Find the intervals on which the function f ( x) = x 3 + 3 x 2 − 9 x is increasing and decreasing. Critical numbers: f ' ( x) = 3x 2 + 6 x − 9 3x 2 + 6 x − 9 = 0 3( x 2 + 2 x − 3) = 0 3( x + 3)( x − 1) = 0 x = {−3,1}
- 9. f ' ( x) = 3x 2 + 6 x − 9 Practice Test an x-value in each interval. Interval (−∞,−3) (−3,1) (1, ∞) Test Value −4 0 2 f ‘(x) f ' (−4) = 15 f ' ( 0) = −9 f ' (2) = 15 f(x) is increasing on (−∞ ,− 3) and (1, ∞) . f(x) is decreasing on (−3,1) .
- 10. The First Derivative Test AP Calculus – Section 3.3
- 11. The First Derivative Test Summary The point where the first derivative changes sign is an extrema.
- 12. The First Derivative Test If c is a critical number of a function f, then: If f ‘(c) changes from negative to positive at c, then f(c) is a relative minimum. If f ‘(c) changes from positive to negative at c, then f(c) is a relative maximum. If f ‘(c) does not change sign at c, then f(c) is neither a relative minimum or maximum. GREAT picture on page 181!
- 13. Visual of First Derivative Test
- 14. Find all intervals of increase/decrease and all relative extrema. f ( x) = x 2 + 8 x + 10 Critical Points: Test: (−∞,−4) f ' ( x) = 2 x + 8 2x + 8 = 0 x = −4 f ' (−5) = 2(−5) + 8 = −2 f is decreasing CONCLUSION: Test: (−4, ∞) f ' ( 0) = 8 f is increasing f is decreasing before -4 and increasing after -4; so f(-4) is a MINIMUM.