Extrema and Average Rates of Change
* A function f is increasing on an interval I if and only if for any two points in I, a positive change in x results in a positive change in f(x)
f(x2)
f(x1)
x1
{
x2
I
* A function f is decreasing on an interval I if and only if for any two points in I, a positive change in x results in a negative change in f(x)
f(x1)
f(x2)
x1
{
x2
I
* A function f is constant on an interval I if and only if for any two points in I, a positive change in x results in a zero change in f(x)
f(x1) = f(x2)
x1
to right
Constant
sin
ea
cr
In
ng
si
ea
cr
∎ increasing from (­∞,­2)
∎ constant from (­2,4)
∎ decreasing from (4, +∞) De
g
As you move from left
the graph of f(x) is:
x2
**Intervals: Since a function is neither increasing nor decreasing at a point the symbols ( and ) are used to describe intervals where a function increases or
decreases**
Describing Increasing/Decreasing/Constant Intervals
is different from describing the Domain intervals of a function
Ex Graph and find the intervals on which the graph is increasing, decreasing, or constant and describe.
f(x) = x3 ­ 3x
click here
from left to right:
Increasing (­∞,-1)
Decreasing (-1, 1)
Increasing (1, +∞)
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Your turn: Graph and determine where the function is increasing, decreasing, or is constant. Support your answers graphically and numerically.
f(x) = x3 ­ 3x
f(x) = 2x2 ­ 8x + 5
f(x) = 3x + 11 if x < ­3.1
1.7 if x > 3.1
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Functions can have critical points. These are the points at which a line drawn tangent to the curve is horizontal or vertical. There are different types of critical points.
I. Extrema ­ Critical points where a function changes its increasing or decreasing
behavior (visually these are peaks or valleys on its graph). At these points the
function has either a maximum or a minimum value.
*Extrema can be Relative or Absolute (there can be several relative max or
min values on a graph, but only one absolute max and/or absolute min value on a graph)
a. Relative Maximum: the greatest value f(x) can attain on some interval of the domain
b. Absolute Maximum: the relative maximum that is the greatest
value f(x) can attain over the entire domain
c. Relative Minimum: the least value f(x) can attain on some interval
of the domain
d. Absolute Minimum: the relative minimum that is the smallest value f(x) can attain over the entire domain
II. Inflection Points ­ places where the graph changes its shape but not its
increasing or decreasing behavior ­ instead the graph changes from being
bent upward to being bent downward or vice versa
inflection point
EX I.
Apply labels in the correct places
abs
min
abs
max
rel
max
rel
min
rel
max
rel
min
rel
max
rel
min
Why is the absolute maximum
where it is?
Why isn't there an absolute
minimum?
*Relative Extrema are also called local extrema *Absolute Extrema are also called global extrema
(Keep in mind that the end behavior can influence weather or not a function has an absolute max or absolute min value)
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Average Rate of Change: the slope of a line between two points on a nonlinear function.
(Remember the slope of any LINEAR function represents a constant rate of change)
For nonlinear functions, the slope changes between different pairs of points, so you can only talk about the average rate of change between any two points.
The average rate of change between any two points on the graph of a function is the slope of the line through those points.
­the line through two points on a curve is called a secant line. The slope of the secant line is denoted msec
­ the average rate of change on the interval [x1, x2] is msec = f(x2) ­ f(x1)
x2 ­ x1
When the average rate of change over an interval is positive, the function increases on average over that interval. When the average rate of change over an interval is negative, the function decreases on average over that interval.
Ex. find the average rate of change of f(x) = ­x3 + 3x on the two intervals a) [­2,­1]
b) [0, 1]
3
3
a.) msec = f(x2) ­ f(x1) = f(­1) ­ f(­2) = [­(­1) + 3(­1)] ­ [­(­2) + 3(­2)] ­1 + 2
x2 ­ x1 ­1 ­ (­2)
= [­(­1) ­ 3] ­ [­(­8) ­ 6]
1
= (1­3) ­(8­6)
= ­2 ­ 2
= ­4
b.)
Average rate of change has many real world applications. One common application involves the average speed of an object traveling over a distance (d) or from a height (h) in a given period of time (t). Because speed is distance traveled per unit time, the average speed of an object cannot be negative.
Ex. If wind resistance is ignored, the distance d(t) in feet an object travels when dropped from a high place is given by d(t) = 16t2 where t is the time in seconds after the object is dropped. Find and interpret the average speed of the object from 2 to 4 seconds.
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