1 / 24

430 likes | 822 Vues

Increasing and Decreasing Functions and the First Derivative Test. Determine the intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema of a function. Standard 4.5a. y. Increasing. Decreasing. Constant. x.

Télécharger la présentation
## Increasing and Decreasing Functions and the First Derivative Test

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Increasing and Decreasing Functions and the First Derivative**Test Determine the intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema of a function Standard 4.5a**y**Increasing Decreasing Constant x**Test for Increasing and Decreasing Functions**Let f be differentiable on the interval (a, b) If f’(x) > 0 then f is increasing on (a, b) If f’(x) < 0 then f is decreasing on (a, b) If f’(x) = 0 then f is constant on (a, b)**Definition of Critical Number**If f is defined at c, then c is a critical number of f if f’(c)=0 or if f’ is undefined at c.**Find the open intervals on which the given function is**increasing or decreasing. 1. Find derivative. 2. Set f’(x) = 0 and solve to find the critical numbers. CRITICAL NUMBERS**Make table to test the sign f’(x) in each interval.**Use the test for increasing/decreasing to decide whether f is increasing or decreasing on each interval.**Find the open intervals on which the given function is**increasing or decreasing.**y**Relative maximum Increasing Decreasing Increasing Relative minimum x**Definition of Relative Extrema**Let f be a function defined at c. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b). f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b). If f(c) is a relative extremum of f, then the relative extremum is said to occur at x = c.**f(c) is a relative maximum of f if there exists an interval**(a, b) containing c such that f(x) ≤ f(c) for all x in (a, b). relative maximum f(c) f(x) f(x) f(x) f(x) f(x) f(x) f(x) f(x) c**2. f(c) is a relative minimum of f if there exists an**interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b). f(x) f(x) f(x) f(x) f(x) f(x) f(c) relative minimum**Occurrence of Relative Extrema**If f has a relative minimum or a relative maximum when x = c, then c is a critical number of f. That is, either f’(c) = 0 or f’(c) is undefined.**First-Derivative Test for Relative Extrema**Let f be continuous on the interval (a, b) in which c is the only critical number. On the interval (a, b) if 1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum. 2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum. 3. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum.**1. f’(x) is negative to the left of x = c and positive to**the right of x = c, then f(c) is a relative minimum. f’(x) is positive f’(x) is negative Relative minimum c**2. f’(x) is positive to the left of x = c and negative**to the right of x = c, then f(c) is a relative maximum. relative maximum f’(x) is positive f’(x) is negative c**f’(x) has the same sign to the left and right of**x = c, then f(c) is not a relative extremum. Not a relative extremum f’(x) is positive f’(x) is positive c**Find all relative extrema of the given function.**Find derivative Set = 0 to find critical numbers CRITICAL NUMBERS**Relative Maximum (-1, 5)**Relative Minimum (1, -3)**Find all relative extrema of the given function.**Relative max: (-2, 0) Relative min: (0, -2)**Relative max:**Relative min:**The graph of f is shown. Sketch a graph of the derivative**of f.**The graph of f is shown. Sketch a graph of the derivative**of f.**The graph of f is shown. Sketch a graph of the derivative**of f.

More Related