Calculus 4'3

1
In the past, one of the important uses of
derivatives was as an aid in curve sketching. We
usually use a calculator or computer to draw
complicated graphs now, but it is still important
to understand the relationships between
derivatives and graphs.
2
First derivative
Curve is rising because the slope is .
Curve is falling because the slope is -.
Possible local maximum or minimum.
Second derivative
Curve is concave up.
Curve is concave down.
Possible inflection point (where concavity
changes).
3
Example
Graph
We can use a chart to organize our thoughts.
First derivative test
This allows you to find the stationary points
negative
positive
positive
4
Example
Graph
First derivative test
5
Example
Graph
NOTE IB will not test specifically on maximum
and minimum problems. They will as increasing
and decreasing questions and your work is
essential!
First derivative test
6
Example
Graph
We then look for inflection points by setting the
second derivative equal to zero.
negative
positive
7
Make a summary table to help you sketch the
graph of the original function.
p
8
You Try

• Find the stationary points on the graph.
• Where is the function increasing?
• Where is the function decreasing?
• Find the local maximum point on the graph.

9
You Try

• Find and describe the nature of the stationary
  points on the graph
• Does the graph have an inflection point?
• Where is the graph increasing?

10
Even though the graphing calculator and the
computer have eliminated the need to routinely
use calculus to graph by hand and to find maximum
and minimum values of functions, we still study
the methods to increase our understanding of
functions and the mathematics involved.
Absolute extreme values are either maximum or
minimum points on a curve.
They are sometimes called global extremes.
They are also sometimes called absolute
extrema. (Extrema is the plural of the Latin
extremum.)
11
Extreme Value Theorem
If f is continuous over a closed interval, then
f has a maximum and minimum value over that
interval.
Maximum minimum at interior points
Maximum minimum at endpoints
Maximum at interior point, minimum at endpoint
12
Local Extreme Values
A local maximum is the maximum value within some
open interval.
A local minimum is the minimum value within some
open interval.
13
Absolute maximum
(also local maximum)
Local maximum
Local minimum
Local minimum
Absolute minimum
(also local minimum)
14
Absolute maximum
(also local maximum)
Local maximum
Local minimum
15
Local Extreme Values
If a function f has a local maximum value or a
local minimum value at an interior point c of its
domain, and if exists at c, then
16
There are no values of x that will make the first
derivative equal to zero.
The first derivative is undefined at x0, so
(0,0) is a critical point.
Because the function is defined over a closed
interval, we also must check the endpoints.
17
To determine if this critical point is actually a
maximum or minimum, we try points on either side,
without passing other critical points.
Since 0lt1, this must be at least a local minimum,
and possibly a global minimum.
18
To determine if this critical point is actually a
maximum or minimum, we try points on either side,
without passing other critical points.
Since 0lt1, this must be at least a local minimum,
and possibly a global minimum.
19
Absolute maximum (3,2.08)
Absolute minimum (0,0)
20
Finding Maximums and Minimums Analytically
21
Example
The function is defined on a closed interval from
-2 to 6. Calculate the maximum value of the
function.