CHAPTER 3 SECTION 3.4 CONCAVITY AND THE SECOND DERIVATIVE TEST

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CHAPTER 3SECTION 3.4CONCAVITY AND THE SECOND
DERIVATIVE TEST
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Definition of Concavity and Figure 3.24
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Sketch 4 graphs a)1 decreasing and concave
upb)1 increasing and concave up, c)1 decreasing
and concave down,d)1 increasing and concave down
y
y
x
x
b
a
y
y
x
x
d
c
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Concave upward
y
y
x
x

• Look at these two graphs. Each is concave
  upward, but one is decreasing and the other is
  increasing. We need to be able to determine
  concavity from the function and not just from the
  graph. For each of the graphs above sketch the
  tangent lines to the graph at a number of
  different points.

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Concave upward
y

• As we move from left to right the slopes of the
  tangent lines are getting less negative. That is
  they are increasing.

x
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Concave upward

• As we move from left to right the slopes of the
  tangent lines are getting larger. That is they
  are increasing.

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When a graph is concave upward

• The slope of the tangent lines are increasing.

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Concave downward
y
y
x
x

• Look at these two graphs. Each is concave
  downward, but one is decreasing and the other is
  increasing. We need to be able to determine
  concavity from the function and not just from the
  graph. For each of the graphs above sketch the
  tangent lines to the graph at a number of
  different points.

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Concave downward

• As we move from left to right, the slopes of the
  tangent lines are getting more negative.
• They are decreasing.

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Concave downward

• As we move from left to right the slopes of the
  tangent lines are getting smaller. That is they
  are decreasing.

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When a graph is concave downward

• The slopes of the tangent lines are decreasing.

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Putting it all together

• For a function f that is differentiable on an
  interval I, the graph of f is
• (i) Concave up on I, if the slope of the tangent
  line is increasing on I or
• (ii) Concave down on I, if the slope of the
  tangent line is decreasing on I

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Linking knowledge

• (i) Concave up on I, if the slope of the tangent
  line is increasing on I.
• If the slope of the tangent line is increasing
  and the slope of the tangent line is represented
  by the first derivative and to determine when
  something is increasing we had to take the
  derivative, then to find where the slope of the
  tangent line (f (x)) is increasing we will need
  to take the derivative of f (x) or find the
  second derivative f (x)

I know, this is a very large run on sentence.
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Linking knowledge

• (ii) Concave down on I, if the slope of the
  tangent line is decreasing on I
• If the slope of the tangent line is decreasing
  and the slope of the tangent line is represented
  by the first derivative and to determine when
  something is decreasing we had to take the
  derivative, then to find where the slope of the
  tangent line (f (x)) is decreasing we will need
  to take the derivative of
• f (x) or find the second derivative f (x)

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Definition of concavity

• For a function f that is differentiable on an
  interval I, the graph of f is
• (i) Concave up on I, if f is increasing on I or
• (ii) Concave down on I, if f is decreasing on I

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Theorem 3.7Test for concavity
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Putting it all together

• Given the function f(x)
• f(x) 0 x-intercepts
• f(x) undefined vertical asymptote
• f(x)gt0 Q-1 or Q-2
• f(x)lt0 Q-3 or Q-4

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Putting it all together
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Putting it all together
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Determining concavity

• Determine the open intervals on which the graph
  is concave upward or concave downward.

• Concavity find second derivative.
• Find hypercritical numbers.
• Set up a chart
• Find concavity

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c 1 c -1 and f is defined on the entire
line
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Setting up the chart
interval Test points Sign of f f concave
(-8, -1) -2 inc upward
(-1,1) 0 - dec downward
(1,8) 2 inc upward
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Points of inflection

• A point of inflection for the graph of f is that
  point where the concavity changes.

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Theorem 3.7 Test for Concavity
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Definition of Point of Inflection and Figure 3.28
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Theorem 3.8 Points of Inflection
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Theorem 3.9 Second Derivative Test and Figure 3.31
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• Example 1 Graph the function f given by
• and find the relative extrema.
• 1st find graph the function.

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• Example 1 (continued)
• 2nd solve f ?(x) 0.
• Thus, x 3 and x 1 are critical values.

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• Example 1 (continued)
• 3rd use the Second Derivative Test with 3 and 1.
• Lastly, find the values of f (x) at 3 and 1.
• So, (3, 14) is a relative maximum and (1, 18)
  is a
• relative minimum.

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Second Derivative Test
If c is a critical number of f(x) and
a. If f(c) gt 0 then ________________________
b. If f(c) lt 0 then ________________________
c. If f(c) 0 or undefined then
__________________________________
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Second Derivative Test
If c is a critical number of f(x) and
(c, f(c)) is a relative min
a. If f(c) gt 0 then ________________________
(c, f(c)) is a relative max
b. If f(c) lt 0 then ________________________
the test fails (use 1st Derivative test)
c. If f(c) 0 or undefined then
__________________________________
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Concave upward
Concave downward
Inflection Points
If f(x) lt 0 ______________
If f(x) 0 ______________
If f(x) gt 0 ______________
If f(x) 0 ______________
If f(x) lt 0 ______________
If f(x) gt 0 ______________
______________
______________
________________
The second derivative gives the same information
about the first derivative that the first
derivative gives about the original function.
For f(x) to increase, _____________
For f(x) to decrease, _____________
For f(x) to increase, _____________
For f(x) to decrease, _____________
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Concave upward
Concave downward
Slopes increase
Slopes decrease
Inflection Points
Where concavity changes
Occur at critical numbers of f(x)
f(x) is constant
f(x) decreases
f(x) increases
If f(x) lt 0 ______________
If f(x) 0 ______________
If f(x) gt 0 ______________
f(x) increases
f(x) is constant
If f(x) 0 ______________
f(x) decreases
If f(x) lt 0 ______________
If f(x) gt 0 ______________
f(x) is conc up
f(x) is conc down
f(x) is a straight line
______________
______________
________________
The second derivative gives the same information
about the first derivative that the first
derivative gives about the original function.
f(x) gt 0
For f(x) to increase, _____________
f(x) lt 0
For f(x) to decrease, _____________
f(x) gt 0
f(x) lt 0
For f(x) to increase, _____________
For f(x) to decrease, _____________
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• Sketch

• Include extrema, inflection points, and intervals
  of concavity.

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• Sketch

No VAs

• Include extrema, inflection points, and intervals
  of concavity.

Critical numbers
None
smooth
Critical numbers
None
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• 1. Find the extrema of

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• 1. Find the extrema of

Crit numbers
2nd Derivative Test
rel max at
rel min at
rel max at
rel min at
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• 2. Sketch

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• 2. Sketch

Crit numbers
2nd Derivative Test
rel max at
rel min at
Crit numbers
Intervals
Test values
f (test pt)
f(x)
Inf pt
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Find a Function

• Describe the function at the point x3 based on
  the following

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Find a Function

• Describe the function at the point x5 based on
  the following

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Find a Function

• Given the function is continuous at the point
  x2, sketch a graph based on the following

(2,3)
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WHY? BECAUSE f(x) is POSITVE!!!!!!!!!!!!!!!
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