mat 1234 calculus i section 3.3 how derivatives affect the shape of a graph (ii)
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MAT 1234Calculus I
Section 3.3
How Derivatives Affect the Shape of a Graph (II)
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HW and ….
WebAssign HW Take time to study for exam 2
The 1st Derv. Test
Find the critical numbers Find the intervals of increasing and
decreasing Determine the local max./min.
The 1st Derv. Test
Find the critical numbers Find the intervals of increasing and
decreasing Determine the local max./min.
Note that intervals of increasing and decreasing are part of the 1st test.
The 2nd Derv. Test
We will talk about intervals of concave up and down
But they are not part of the 2nd test.
Preview
We know the critical numbers give the potential local max/min.
How to determine which one is local max/min?
Preview
We know the critical numbers give the potential local max/min.
How to determine which one is local max/min?
30 second summary!
Preview
Concave Up
𝑓 ’ (𝑐)=0
Concave Down
𝑓 ’ (𝑐)=0
Preview
We know the critical numbers give the potential local max/min.
How to determine which one is local max/min?
30 second summary! We are going to develop the theory
carefully so that it works for all the functions that we are interested in.
Preview
Part I Increasing/Decreasing Test The First Derivative Test
Part II Concavity Test The Second Derivative Test
Definition
(a) A function is called concave upward on an interval if the graph of lies above all of its tangents on .
(b) A function is called concave downward on an interval if the graph of lies below all of its tangents on .
Concavity
is concave up on
Potential local min.
Concavity
is concave down on
Potential local max.
Concavity
has no local max. or min.
has an inflection point at
c
Concave down
Concave up
Definition
An inflection point is a point where the concavity changes (from up to down or from down to up)
Concavity Test
(a) If on an interval , then is concave upward on .
(b) If on an interval , then is concave downward on .
Concavity Test
(a) If on an interval , then is concave upward on .
(b) If on an interval , then is concave downward on .
Why?
Why?
implies is increasing. i.e. the slope of tangent lines is increasing.
( ) ( )d
f x f xdx
Why?
implies is decreasing. i.e. the slope of tangent lines is decreasing.
Example 3
Find the intervals of concavity and the inflection points
1362)( 23 xxxxf
Example 31362)( 23 xxxxf
(a) Find ,
and the values of such that
)(xf )(xf
x 0)( xf
Example 31362)( 23 xxxxf
(b) Sketch a diagram of the subintervals formed by the values found in part (a). Make sure you label the subintervals.
Example 31362)( 23 xxxxf
(c) Find the intervals of concavity and inflection point(s).
has an inflection point at ( , )
Expectation
Answer in full sentence. The inflection point should be given by
the point notation.
Example 3 Verification
The Second Derivative Test
Suppose is continuous near .
(a) If and , then has a local minimum at .(b) If and , then f has a local maximum at c.
(c) If , then no conclusion (use 1st derivative test)
Second Derivative Test
Suppose
If then has a local min. at
0)( cf
0)( cf
𝑐
𝑓 ”(𝑐)>0
𝑓 ’ (𝑐)=0
Second Derivative Test
Suppose
If then has a local max. at
0)( cf
0)( cf
𝑐
𝑓 ”(𝑐)<0
𝑓 ’ (𝑐)=0
Example 4 (Example 2 Revisit)
Use the second derivative test to find the local max. and local min.
10249)( 23 xxxxf
Example 4 (Example 2 Revisit)
(a) Find the critical numbers of
10249)( 23 xxxxf
Example 4 (Example 2 Revisit)
(b) Use the Second Derivative Test to find the local max/min of
10249)( 23 xxxxf
The local max. value of is
The local min. value of is
Second Derivative Test
Step 1: Find the critical points Step 2: For each critical point,
• determine the sign of the second derivative;• Find the function value• Make a formal conclusion
Note that no other steps are required such as finding intervals of inc/dec, concave up/down.
The Second Derivative Test
(c) If , then no conclusion
The Second Derivative Test
(c) If , then no conclusion
4
3
2
2
( )
( ) 4 0
0
( ) 12
(0) 12 0 0
f x x
f x x
x
f x x
f
The Second Derivative Test
(c) If , then no conclusion
4
3
2
2
( )
( ) 4 0
0
( ) 12
(0) 12 0 0
g x x
g x x
x
g x x
g
The Second Derivative Test
(c) If , then no conclusion
3
2
( )
( ) 3 0
0
( ) 6
(0) 6 0 0
h x x
h x x
x
h x x
h
The Second Derivative Test
Suppose is continuous near .
(a) If and , then has a local minimum at .(b) If and , then f has a local maximum at c.
(c) If , then no conclusion (use 1st derivative test)
Which Test is Easier?
First Derivative Test Second Derivative Test
Final Reminder
You need intervals of increasing/decreasing for the First Derivative Test.
You do not need intervals of concavity for the Second Derivative Test.
Classwork
Do part (a), (d) and (e) only