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Recaping Last Lectures

The document discusses increasing and decreasing functions. It defines an increasing function as one where f(x1) < f(x2) when x1 < x2, and a decreasing function as one where f(x1) > f(x2) when x1 < x2. It then states that if the derivative f'(x) is positive, the function is increasing, and if f'(x) is negative, the function is decreasing. It provides examples of analyzing functions to determine the intervals where they are increasing or decreasing based on the sign of the derivative.

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Ghulam Fareed
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43 views31 pages

Recaping Last Lectures

The document discusses increasing and decreasing functions. It defines an increasing function as one where f(x1) < f(x2) when x1 < x2, and a decreasing function as one where f(x1) > f(x2) when x1 < x2. It then states that if the derivative f'(x) is positive, the function is increasing, and if f'(x) is negative, the function is decreasing. It provides examples of analyzing functions to determine the intervals where they are increasing or decreasing based on the sign of the derivative.

Uploaded by

Ghulam Fareed
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Increasing and Decreasing

Functions
Lesson 5.1
The Ups and Downs
Think of a function as a roller coaster going
from left to right

Uphill
• Slope > 0
• Increasing
function
Downhill
• Slope < 0
• Decreasing function
2
Definitions
Given function f defined on an interval
• For any two numbers x1 and x2 on the interval
Increasing function
• f(x1) < f(x2) when x1 < x2

Decreasing function X1 X2 X1 X2 f(x)

• f(x1) > f(x2) when x1< x2

3
Increasing/Decreasing and the
Derivative
Assuming existence of derivative on
interval

If f '(x) > 0 for each x


• f(x) increasing on
interval
If f '(x) < 0 for each x
• f(x) decreasing on
interval What if f '(x) = 0 on the interval?
What could you say about f(x)?
4
Check These Functions
By graphing on calculator, determine the
intervals where these functions are
• Increasing
• Decreasing
2 3
f ( x)  x  x 2  4 x  2
3

y  2 x  5
x3
f ( x) 
x4

5
Critical Numbers
Definition
Numbers c in the domain of f where
• f '(c) = 0

• f '(c) does not exist


Critical Points
2
x
f ( x) 
x2

6
Applying Derivative Test
Given a function f(x)
Determine the derivative f '(x)
Find critical points …
• Where f '(x) = 0 or f '(x) does not exist
Evaluate derivative between or on either
side of the critical points
2 3
Try it with this function f ( x)  x  x 2  4 x  2
3
7
Test for Increasing and Decreasing
Functions

 Let f be a function that is continuous on the closed


interval [a, b] and differentiable on the open interval (a,
b).

 1. If f’(x) > 0 (positive) for all x in (a, b), then f is


increasing on [a, b].

 2. If f’(x) < 0 (negative) for all x in (a, b), then f is


decreasing on [a, b].

 3. If f’(x) = 0 for all x in (a, b), then f is constant on [a,


b]. (It is a critical number.)

8
Determining intervals on which f is
increasing or decreasing

Example:
Find the open intervals on which

3 2
f ( x)  x  x
3

is increasing or decreasing.

9
Guidelines for finding intervals
Let f be continuous on the interval (a, b). To
find the open intervals on which f is
increasing or decreasing, use the following
steps:
(1) Locate the critical number of f in (a, b),
and use these numbers to determine test
intervals
(2) Determine the sign of f’(x) at one test
value in each of the intervals
(3) If f’(x) is positive, the function is
increasing in that interval. If f’(x) is negative,
the function is decreasing in that interval. 10
The First Derivative Test
Let c be a critical number of a function f that is
continuous on an open interval I containing c. If f is
differentiable on the interval, except possibly at c,
then f(c) can be classified as follows.
(1) If f’(x) changes from negative to positive at c,
then f(c) is a relative minimum of f.
(2) If f’(x) changes from positive to negative at c,
then f(c) is a relative maximum of f.
(3) If f’(x) does not change signs at c, then f(c) is
neither a relative maximum nor a minimum.

11
First Derivative Test

More Examples

12
13
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