MAT 1221 Survey of Calculus
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MAT 1221 Survey of Calculus Section 3.3 Concavity and the Second Derivative Test http://myhome.spu.edu/lauw
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MAT 1221 Survey of Calculus. Section 3.3 Concavity and the Second Derivative Test. http://myhome.spu.edu/lauw. Expectations. Check your algebra. Check your calculator works Formally answer the question with the expected information. 1 Minute…. - PowerPoint PPT Presentation
Transcript of MAT 1221 Survey of Calculus
MAT 1221Survey of Calculus
Section 3.3Concavity and the Second
Derivative Test
http://myhome.spu.edu/lauw
Expectations Check your algebra. Check your calculator works Formally answer the question with the
expected information
1 Minute… You can learn all the important concepts
in 1 minute.
1 Minute… Critical numbers – give the potential local
max/mins
1 Minute… Critical numbers – give the potential local
max/mins
If the graph is “concave down” at a critical number, it has a local max
1 Minute… Critical numbers – give the potential local
max/mins
If the graph is “concave up” at a critical number, it has a local min
1 Minute… You can learn all the important concepts
in 1 minute. We are going to develop the theory
carefully so that it works for all the functions that we are interested in.
There are a few definitions…
Preview Define
• Second Derivative• Concavities
Find the intervals of concave up and concave down
The Second Derivative Test
Second Derivative
Second Derivative 5 32f x x x
ddx
4 25 6f x x x
320 12f x x x
ddx
Given a function
which is a function.
)( of derivativefirst the)( of derivative the)(xf
xfxf
Higher Derivatives
Given a function
)( of derivative second the)( of derivative the
)()(
xfxf
xfdxdxf
Higher Derivatives
Concave Up(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
yConcave
down
Concave up
xc
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 f is concave downward on .
Concavity Test
(a) If on an interval , then is concave upward on .(b) If on an interval , then f is concave downward on .
Why? (Hint: ) ( ) ( )df x f xdx
Why? implies is increasing. i.e. the slope of tangent lines is increasing.
( ) ( )df x f xdx
Why? implies is decreasing. i.e. the slope of tangent lines is decreasing.
( ) ( )df x f xdx
Example 1Find the intervals of concavity and the inflection points
1362)( 23 xxxxf
Example 11362)( 23 xxxxf
1. Find , and the values of such that
)(xf )(xf
x 0)( xf
Example 11362)( 23 xxxxf
2. Sketch a diagram of the subintervals formed by the values found in step 1. Make sure you label the subintervals.
Example 11362)( 23 xxxxf
3. Find the intervals of concavity and inflection point.
1 8f
Example 11362)( 23 xxxxf
The Second Derivative TestSuppose is continuous near .(a) If and , then has a local minimum at c.(b) If and , then f has a local maximum at .
(c) If , then no conclusion (use 1st derivative test)
Second Derivative TestSupposeIf then has a local min at
0)( cf0)( cf
c
𝑓 ”(𝑐)>0
𝑓 ’ (𝑐)=0
x
y
Second Derivative TestSupposeIf then has a local max at 0)( cf
0)( cf
c
𝑓 ”(𝑐)<0
𝑓 ’ (𝑐)=0
x
y
The Second Derivative Test(c) If , then no conclusion
The Second Derivative TestIf , then no conclusion
4
3
2
2
( )
( ) 4 0 0
( ) 12
(0) 12 0 0
f x x
f x xx
f x x
f
The Second Derivative TestIf , then no conclusion
4
3
2
2
( )
( ) 4 0 0
( ) 12
(0) 12 0 0
g x x
g x xx
g x x
g
The Second Derivative TestIf , then no conclusion
3
2
( )
( ) 3 0 0
( ) 6(0) 6 0 0
h x x
h x xx
h x xh
The Second Derivative TestSuppose is continuous near .(a) If and , then has a local minimum at c.(b) If and , then f has a local maximum at .
(c) If , then no conclusion (use 1st derivative test)
Example 2Use the second derivative test to find the local max. and local min.
10249)( 23 xxxxf
Example 2(a) Find the critical numbers of
10249)( 23 xxxxf
Example 2(b) Use the Second Derivative Test to find the local max/min of
10249)( 23 xxxxf
The local max. value of isThe local min. value of is
2 10, 4 6f f
Review Example 1 & 2 illustrate two different but
related problems. 1. Find the intervals of concavity and
inflection points. 2. Find the local max. /min. values
Expectations Follow the steps to solve the two
problems
