mat 1234 calculus i
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MAT 1234 Calculus I. Section 3.3 How Derivatives Affect the Shape of a Graph (II). http://myhome.spu.edu/lauw. Next. Wednesday Quiz: 3.3,3.4 Exam II: Next Thursday. Preview. We know the critical numbers give the potential local max/min. How to determine which one is local max/min?. - PowerPoint PPT PresentationTRANSCRIPT
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MAT 1234Calculus I
Section 3.3How Derivatives Affect the
Shape of a Graph (II)
http://myhome.spu.edu/lauw
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HW and …. WebAssign HW Take time to study for exam 2
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The 1st Derv. Test Find the critical numbers Find the intervals of increasing and
decreasing Determine the local max./min.
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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.
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The 2nd Derv. Test We will talk about intervals of concave
up and down But they are not part of the 2nd test.
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Preview We know the critical numbers give the
potential local max/min. How to determine which one is local
max/min?
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Preview We know the critical numbers give the
potential local max/min. How to determine which one is local
max/min? 30 second summary!
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Preview
Concave Up
𝑓 ’ (𝑐)=0
Concave Down
𝑓 ’ (𝑐)=0
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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.
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PreviewPart I Increasing/Decreasing Test The First Derivative TestPart II Concavity Test The Second Derivative Test
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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 .
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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
c
Concave down
Concave up
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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 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 is concave downward on .
Why?
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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.
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Example 3Find the intervals of concavity and the inflection points
1362)( 23 xxxxf
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Example 31362)( 23 xxxxf
(a) Find , and the values of such that
)(xf )(xf
x 0)( xf
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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.
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Example 31362)( 23 xxxxf
(c) Find the intervals of concavity and inflection point(s).
has an inflection point at ( , )
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Expectation Answer in full sentence. The inflection point should be given by
the point notation.
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Example 3 Verification
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The Second Derivative TestSuppose 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)
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Second Derivative TestSuppose
If then has a local min. at
0)( cf
0)( cf
𝑐
𝑓 ”(𝑐)>0
𝑓 ’ (𝑐)=0
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Second Derivative TestSuppose
If then has a local max. at
0)( cf
0)( cf
𝑐
𝑓 ”(𝑐)<0
𝑓 ’ (𝑐)=0
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Example 4 (Example 2 Revisit)Use the second derivative test to find the local max. and local min.
10249)( 23 xxxxf
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Example 4 (Example 2 Revisit)(a) Find the critical numbers of
10249)( 23 xxxxf
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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 isThe local min. value of is
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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.
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The Second Derivative Test(c) If , then no conclusion
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The Second Derivative Test(c) If , then no conclusion
4
3
2
2
( )
( ) 4 00
( ) 12
(0) 12 0 0
f x x
f x xx
f x x
f
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The Second Derivative Test(c) If , then no conclusion
4
3
2
2
( )
( ) 4 00
( ) 12
(0) 12 0 0
g x x
g x xx
g x x
g
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The Second Derivative Test(c) If , then no conclusion
3
2
( )
( ) 3 00
( ) 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 .(b) If and , then f has a local maximum at c.
(c) If , then no conclusion (use 1st derivative test)
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Which Test is Easier? First Derivative Test Second Derivative Test
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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.
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Classwork Do part (a), (d) and (e) only