unit 4 applications of derivatives. slide 2 4.1 critical number test types of maximums and minimums...
TRANSCRIPT
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Unit 4
Applications of Derivatives
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Slide 2
4.1 Critical Number Test
Types of Maximums and Minimums
Absolute/Global Extreme Values• Maximum – • Minimum –
Local/Relative Extreme Values• Maximum –
• Minimum –
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Slide 3
4.1 Critical Number Test
Critical Number/Value –
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Slide 4
4.1 Critical Number Test1) For the following picture, find the absolute extrema, local extrema and critical numbers.
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Slide 5
4.1 Critical Number TestExtreme Value Theorem – If f is continuous on a closed interval [a, b], then f has on that interval.
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Slide 6
4.1 Critical Number Test
Critical Numbers Test for Finding Extrema
If f is continuous on a closed interval [a, b], then it’s absolute extreme values are paired with critical numbers or endpoints.
Steps for finding the absolute extrema.1.
2.
3.
4.
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Slide 7
4.1 Critical Number TestFind the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f.
1) 3 212 5 on 2, 2
2f x x x x
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Slide 8
4.1 Critical Number TestFind the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f.
2) 3 1 on 1, 1f x x x
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Slide 9
4.2 Mean Value Theorem
If ( ) is continuous at every point of the closed interval , and
differentiable at every point of its interior , , then there is at least
( ) - ( )one point in , at which '( ) .
-
y f x a b
a b
fb f ac a b f c
b a
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Slide 10
4.2 Mean Value TheoremDetermine if f satisfies the conditions of the mean value theorem. If so, find all possible values of c.
1) 3 3 5 on 3, 2f x x x
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Slide 11
4.3 First Derivative Test
1 2If is defined for all in some interval and let and be in , then 1. is increasing if and
f x I x x I
f
2. is decreasing if and 3.
f
is constant if and
f
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Slide 12
4.3 First Derivative Test
Suppose is continuous on , and differentiable on ,
1. If ' 0 for all in , , then
2. If ' 0 for all in , , then
f a b a b
f x x a b
f x x a b
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Slide 13
4.3 First Derivative Test
First Derivative Test (for finding local extrema)
Let be continuous on , and
1. If ' 0 to the left of and ' 0 to the right of ,
then is a
f a b a c b
f x c f x c
f c
2. If ' 0 to the left of and ' 0 to the right of ,
then is a
f x c f x c
f c
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Slide 14
4.3 First Derivative Test
2 2
Find the intervals over which is increasing and decreasing. Give the values of for which has local extreme values.1) 5 5 2
fx f
f x x x
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Slide 15
4.3 First Derivative Test
2
Find the intervals over which is increasing and decreasing. Give the values of for which has local extreme values.1) sin sin on 0, 2
fx f
f x x x
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Slide 16
4.4 Second Derivative Test• If the second derivative is positive, a curve looks like
• If the second derivative is negative, a curve looks like
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Slide 17
4.4 Second Derivative Test• f is concave up on (a, b) if and only if the slopes of the tangents
to f are increasing• which means
• which means
• f is concave down on (a, b) if and only if the slopes of the tangents
to f are decreasing• which means
• which means
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Slide 18
4.4 Second Derivative Test(c, f(c)) is an inflection point if and only if
1.
2.
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Slide 19
4.4 Second Derivative Test
Second Derivative Test (for finding some local extrema)
1. If ' 0 and '' 0, then
2. If ' 0 and '' 0, then
f c f c
f c f c
3. If ' 0 and '' 0, then f c f c
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Slide 20
4.4 Second Derivative Test
5 3Find the local extreme values using the second derivative test.1) 3 5 1f x x x
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Slide 21
Relationship of f’ and f’’ to ff
4.4 Second Derivative Test
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Slide 22
4.4 Second Derivative Test
2
3
Find the intervals for which is increasing and decreasing, local extreme values, intervals for which is concave up and concave down, inflection points and provide a sketch of the graph.1)
ff
f x x 1 x