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Lesson Plan #23
Class: AP Calculus
Date: Tuesday November 1st, 2011
Topic: Concavity
Aim: How do we determine the concavity of a function?
Objectives:
1) Students will be able to determine the concavity of a function.
HW# 23:
For the function 𝑦 = 𝑥 4 − 3𝑥 2 + 2, find the critical points, inflection points, the absolute minimum value of 𝑦, and
relative maximum points.
Do Now:
1) Suppose f ' ( x)  x( x  2) 2 ( x  3) . Which of the following are true?
I. f has a local (relative) maximum at x  3
II. f has a local minimum at x  0
III. f neither a local maximum nor a local minimum at x  2
A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III
2) Find the minimum value of the function 𝑦 = −4√2 − 𝑥
3) A) Find the relative extrema of the function
f ( x) 
1 3
x x
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B) Draw a rough sketch of the function
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Where do you see the graph facing up?
Where do you see the graph facing down?
At what point does the graph change from facing up to facing down?
Definition of Concavity:
Let f be a differentiable function on an open interval I . The graph of
on the interval and concave down on I if
f is concave up on I if f ' is increasing
f ' is decreasing on the interval.
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Test for concavity:
Let f be a function whose second derivative exists on an open interval I.
f "( x)  0 for all x in I , then the graph of f is concave up in I
2) If f "( x)  0 for all x in I , then the graph of f is concave down in I
1) If
The point at which concavity changes is called an inflection point.
If
 c, f (c)  is a point of inflection of the graph of
f , then either f "(c)  0 or f "(c) is undefined.
Exercises:
Find the open intervals where the graph is concave up and those on which it is concave
down and find the coordinates of the point(s) of inflection, if any
1)
y  x2  x  2
2)
y   x3  3x 2  2
3)
f ( x)  x 4  4 x 3
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2nd Derivative Test:
Let f be a function such that
f ' (c)  0 and the second derivative of f exists on an open interval containing c .
1) If f " (c)  0 , then f (c ) is a relative minimum.
2) If f " (c)  0 , then f (c ) is a relative maximum
If f " (c)  0 , the test fails. In such cases, you can use the First derivative test.
Use the 2nd Derivative test to find the relative extrema for
1)
f ( x)  x 3  3 x 2  3
2)
f ( x)  x 4  4 x 3  2
Find all relative extrema and points of inflection
1)
f ( x)  2 x 4  8 x  3
If enough time:
Sample Test Questions:
1) The function f ( x)  x  4 x has
A) One relative minimum and two relative maxima
C) two relative maxima and no relative minimum
E) two relative minima and one relative maximum
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B) One relative minimum and one relative maximum
D) two relative minima and no relative minimum
2) For the above question, what is the number of inflection points?
A) 0
B) 1
C) 2
D) 3
E) 4
3) What is the maximum value of the function
A) 1
B) 0
C) 3
D) 6
f ( x)  x  4 x  6 on the closed interval 1,4 ?
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E) None of the other choices
x , f ' ( x)  ( x  2) 4 ( x  1)3 , it follows that the function f has
A) a relative minimum at x  1
B) a relative maximum at x  1
C) both a relative minimum at x  1and a relative maximum at x  2
4) If, for all
D) neither a relative maximum nor a relative minimum
E) relative minima at x  1and at x  2
5) The number of inflection points of f ( x)  3x  10 x is
A) 4
B) 3
C) 2
D) 1
E) 0
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6) The minimum value of
A)
1
2
B) 1
f ( x)  x 2 
C) 3
3
2
1
 x  2 is
on the interval
x
2
D) 4.5
E) 5
f ( x)  x 4  4 x3  4 x 2  6 increase?
B) x  2 only
C) 0  x  1and x  2
E) 1  x  2 only
7) On which interval(s) does the function
A)
D)
x  0 and 1  x  2
0  x  1only
8) The graph of 𝑦 = 𝑥 3 + 21𝑥 2 − 𝑥 + 1is concave down for
A) 𝑥 < 7
B) −7 < 𝑥 < 7
C) all 𝑥
D) 𝑥 > 7
E) 𝑥 < −7 𝑎𝑛𝑑 𝑥 > 7
9) Let 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 3𝑥 + 7. Which of the following statement is true? (Try doing this with the aid of your calculator)
A) 𝑓 has a relative extremum at 𝑥 = 1 and no inflection points.
B) 𝑓is everywhere increasing and does not change concavity
C) 𝑓has no relative extrema but has an inflection point at 𝑥 = 1
D) 𝑓has a relative maximum and an inflection point at 𝑥 = 1
E) 𝑓has a relative minimum and an inflection point at 𝑥 = 1
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