Post on 16-Dec-2015
MAT 1221Survey of Calculus
Section 3.1
Increasing and Decreasing Functions
http://myhome.spu.edu/lauw
HW
WebAssign HW 3.1 Additional HW listed at the end of the
handout (need to finish, but no need to turn in)
Please do your HW ASAP. Do not wait for Tuesday to do your HW.
Read 3.2 for tomorrow.
Tutoring Bonus-Bonus Points
5 points for tutoring in the tutoring room 5 additional points from Caylee
Max/Min
We are interested in max/min values• Minimize the production cost• Maximize the profit• Maximize the sunlight exposure of plants
Preview
Define increasing/Decreasing Function Increasing/Decreasing Test
Increasing/ Decreasing Functions
(a) A function is increasing on an interval if for any two numbers in the interval implies
x
y
t s
)(xfy
Increasing/ Decreasing Test
(a) If on an interval, then is increasing on that interval.
(b) If on an interval, then is decreasing on that interval.
0)( xf
0)( xf
x
Increasing/ Decreasing Testy is increasing on
a b
)(xfy
x
Increasing/ Decreasing Testy is increasing on
on
a b
)(xfy
x
Increasing/ Decreasing Testy
is decreasing on
a b
)(xfy
x
Increasing/ Decreasing Testy
is decreasing on on
a b
)(xfy
Increasing/ Decreasing Test
How to find the intervals of increasing and decreasing?
x
y
a c
)(xfy
b
>0
=0
<0
Critical Number
A critical number of a function is a number c in the domain of such that either or does not exist.0)( cf )(cf
Critical Number
A critical number of a function is a number c in the domain of such that either or does not exist.
For “nice” functions such as polynomials, critical numbers are those with
0)( cf )(cf
0)( cf
Increasing/ Decreasing Test
To find the intervals of increasing and decreasing, we look for the values of x such that
(The critical numbers)
x
y
a c
)(xfy
b
>0
=0
<0
Increasing/ Decreasing Test
The critical numbers divided the interval into subintervals.
xa cb
),( a ),( ba ),( cb ),( c
0)( xf 0)( xf0)( xf
x
Increasing/ Decreasing Test
The critical numbers divided the interval into subintervals.
On each subinterval, the signs of the are the same (Why?).
a cb
),( a ),( ba ),( cb ),( c
0)( xf 0)( xf0)( xf
Example 1
Find the intervals of increasing and decreasing of
106)( 2 xxxf
Example 1106)( 2 xxxf
1. Find the critical numbers
Example 1106)( 2 xxxf
2. Sketch a diagram of the subintervals formed by the critical numbers
Example 1106)( 2 xxxf
3. For each subinterval, pick a point and compute .
Expected Solution Steps
1. Find the critical numbers
2. Sketch a diagram of the subintervals formed by the critical numbers
3. For each subinterval, pick a point and compute
Test =…=-999<0
So, <0 on .
Thus, is decreasing on .