MAT 1221 Survey of Calculus
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Transcript of MAT 1221 Survey of Calculus
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MAT 1221Survey of Calculus
Section 3.3Concavity and the Second
Derivative Test
http://myhome.spu.edu/lauw
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Expectations Check your algebra. Check your calculator works Formally answer the question with the
expected information
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1 Minute… You can learn all the important concepts
in 1 minute.
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1 Minute… Critical numbers – give the potential local
max/mins
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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
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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
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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…
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Preview Define
• Second Derivative• Concavities
Find the intervals of concave up and concave down
The Second Derivative Test
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Second Derivative
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Second Derivative 5 32f x x x
ddx
4 25 6f x x x
320 12f x x x
ddx
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Given a function
which is a function.
)( of derivativefirst the)( of derivative the)(xf
xfxf
Higher Derivatives
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Given a function
)( of derivative second the)( of derivative the
)()(
xfxf
xfdxdxf
Higher Derivatives
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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 .
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Concavity is concave up on
Potential local min.
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Concavity is concave down on
Potential local max.
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Concavity
has no local max. or min. has an inflection point at
yConcave
down
Concave up
xc
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Definition An inflection point is a point where the
concavity changes (from up to down or from down to up)
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Concavity Test
(a) If on an interval , then is concave upward on .(b) If on an interval , then f is concave downward on .
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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
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Why? implies is increasing. i.e. the slope of tangent lines is increasing.
( ) ( )df x f xdx
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Why? implies is decreasing. i.e. the slope of tangent lines is decreasing.
( ) ( )df x f xdx
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Example 1Find the intervals of concavity and the inflection points
1362)( 23 xxxxf
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Example 11362)( 23 xxxxf
1. Find , and the values of such that
)(xf )(xf
x 0)( xf
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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.
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Example 11362)( 23 xxxxf
3. Find the intervals of concavity and inflection point.
1 8f
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Example 11362)( 23 xxxxf
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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)
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Second Derivative TestSupposeIf then has a local min at
0)( cf0)( cf
c
𝑓 ”(𝑐)>0
𝑓 ’ (𝑐)=0
x
y
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Second Derivative TestSupposeIf then has a local max at 0)( cf
0)( cf
c
𝑓 ”(𝑐)<0
𝑓 ’ (𝑐)=0
x
y
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The Second Derivative Test(c) If , then no conclusion
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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
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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
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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
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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)
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Example 2Use the second derivative test to find the local max. and local min.
10249)( 23 xxxxf
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Example 2(a) Find the critical numbers of
10249)( 23 xxxxf
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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
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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
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Expectations Follow the steps to solve the two
problems