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Published byAlban Clarence Stanley Modified over 9 years ago
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Problem of the Day (Calculator allowed)
Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5, and f(9) = -5, which of the following must be true? I. f has at least 2 zeroes II. The graph of f has at least one horizontal tangent. III. For some c, 2 < c < 5, f(c) = 3. A) None B) I only C) I and II only D) I and III only E) I, II, and III
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Problem of the Day (Calculator allowed)
Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5, and f(9) = -5, which of the following must be true? I. f has at least 2 zeroes II. The graph of f has at least one horizontal tangent. III. For some c, 2 < c < 5, f(c) = 3. A) None B) I only C) I and II only D) I and III only E) I, II, and III I. 2 sign changes implies 2 zeroes II. Rolle's Theorem III. Intermediate Value Theorem
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3-4: Concavity & The Second Derivative Test
Objectives: Discuss concavity as an indicator of function behavior Recognize inflection as a change in the rate of change Use the 2nd Derivative Test ©2002 Roy L. Gover (
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Definition Concave up means the graph of f is above the tangent lines. f
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Definition Concave down means the graph of f is below the tangent lines. f
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Definition Concave up “holds water”
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Definition Concave down “spills water”
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Analysis Is there a relationship between the graphs of f(x) & f ’(x)?
Is there a relationship between the concavity of f(x) and f’(x)? Is there a relationship between where concavity changes and f’(x)? Where does concavity change?
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Definition f’(x) increasing
Graph of f is concave up on interval I if f’ is increasing on I f’(x) increasing Concave Up
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Definition f’(x) decreasing
Graph of f is concave down on interval I if f’ is decreasing on I Concave Down f’(x) decreasing
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f’ >0: slope of the tangent lines are positive; f is incr.
Review f’ >0: slope of the tangent lines are positive; f is incr. f’ <0: slope of the tangent lines are negative; f is decr. f’ =0: slope of the tangent line is zero;f is neither increasing nor decreasing.
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Important Idea f ’’>0:slope of tangent lines are becoming more positive (less negative) from left to right. f(x) f ‘(x) f”(x)
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Important Ideas f(x) f ’’=0:slope of tangent lines are not changing. f ‘(x) f”(x)
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Important Ideas f(x) f ’’<0:slope of tangent lines are becoming more negative (less positive) from left to right. f ‘(x) f”(x)
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Important Idea Let f be a function such that f” exists on (a,b), then:
f” (x)>0 for all x in (a,b) f is concave up. f” (x)<0 for all x in (a,b) f is concave down.
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Procedure Determining Concavity:
1. Locate x values at which f ’’=0 or undefined. 2. Use these x values to determine intervals. 3. Test the sign of f ’’ in each interval
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Example Determine the open intervals on which
is concave up and concave down...
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Example Step 1:Find the values of x where f ” =0 or undefined
Step 2:Make a table using intervals determined in step 1 Step 3:Choose a value in each interval & evaluate the 2nd derivative at the value
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Warm-Up Determine the open intervals on which the graph of
is concave up and concave down.
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Solution f(x) Up Down Interval Test Value -2 2 Sign f ”(x) + - Concl.
2 Sign f ”(x) + - Concl. Up Down
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Definition Inflection point is the point where concavity changes.
Inflection points occur where f’’(x)=0 or is undefined but f ”(x)=0 or undefined doesn’t guarantee an inflection point.
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Important Idea An inflection point is the point where concavity changes. An inflection point is where the rate of change changes from increasing to decreasing or vice versa.
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Try This Confirm that has a point of inflection at (0,0).
No inflection point at (0,0)
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Definition 2nd Derivative Test:Let f be a function such that f’(c)=0 and f” exists: If f’’(c)>0, then f(c) is a local min If f’’ (c)<0, then f(c) is a local max If f” (c)=0, test fails
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Procedure Second Derivative Test:
1. Find critical numbers by setting f’(x)=0. 2.Find f’’(c) where c is a critical number. 3. f ”(c)>0 local min; f ”(c)<0 local max.
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Example Find the relative extrema (max and/or min) of:
using the second derivative test
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Try This Find the relative extrema (max and/or min) of:
using the second derivative test
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Solution Max at (1,2) What did you do to determine there was no extrema at (0,0) since f ”(0)=0? Min at (-1,-2)
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Lesson Close How do you test for concavity?
To test for local extrema, do you prefer the 1st derivative or 2nd derivative test? Why?
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Assignment 195/1-5 odd, odd
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