Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Ap calculus speeding problem by Ron Eick 1215 views
- Ap calc 8.28.15 by Ron Eick 259 views
- Ap calc warmup 9.4.14 by Ron Eick 418 views
- Calculus AB - Slope of secant and t... by Kenyon Hundley 58668 views
- Lesson 20: The Mean Value Theorem by Matthew Leingang 1132 views
- Angel tarot and intuition workshop by Ron Eick 404 views

0 views

Published on

No Downloads

Total views

0

On SlideShare

0

From Embeds

0

Number of Embeds

0

Shares

0

Downloads

24

Comments

0

Likes

2

No embeds

No notes for slide

- 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.

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment