Transcript Extremas – First Derivative Test & Second Derivative Test

Calculus - Santowski
Calculus - Santowski
7/6/2015
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1. Determine the first and second derivatives of
the function f(x) = x4 – 4x3 + 4x2
2. Determine the first and second derivatives of
the function f(x) = x4 – 4x3
3. Determine the first and second derivatives of
the function f(x) = x2e-x
4. Determine the first and second derivatives of
the function f(x) = 2sin(x) + sin2(x)
Calculus - Santowski
7/6/2015
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1. Explain what the First Derivative Test is
and why it “works”
2. Explain what the Second Derivative Test is
and why it “works”
3. Work with the FDT & SDT to classify
extrema
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Let f be a differentiable function with f '(c) = 0
then
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1. If f '(x) changes from positive to negative, then
f has a relative maximum at c.
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2. If f '(x) changes from negative to positive, then
f has a relative minimum at c.
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3. If f '(x) does not change sign at c, then f has
neither an maximum or minimum at c.
(Stationary point)
Calculus - Santowski
7/6/2015
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Your task: Write an
explanation/clarification/rationalization of
the FDT. Include diagrams in your
explanation
Calculus - Santowski
7/6/2015
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Use the FDT to classify all extrema of the
function f(x) = x4 – 4x3 + 4x2
(NOTE: Sign charts are NOT allowed!)
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7/6/2015
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Let f be a twice differentiable function near
x = c such that f '(c) = 0. Then
1. If f ''(x) > 0 then f(c) is a relative minimum.
2. If f ''(x) < 0 then f(c) is a relative maximum.
3. If f ''(x) = 0 then use the first derivative
test to classify the extrema.
Calculus - Santowski
7/6/2015
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Your task: Write an
explanation/clarification/rationalization of
the FDT. Include diagrams in your
explanation
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7/6/2015
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Use the function f(x) = x4 – 4x3 to show
algebraically HOW the SDT can be used to
classify the extremas as either (i) maximums,
(ii) minimums, or (iii) stationary points
Calculus - Santowski
7/6/2015
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ex 1. Find and classify all extrema using
FDT of f(x) = 3x5 - 25x3 + 60x.
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ex 2. Find and classify all extrema using
SDT of f(x) = 3x4 - 16x3 + 18x2 + 2.
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Ex 3. Find the intervals of increase and
decrease and max/min values of f(x) =
cos(x) – sin(x) on (-,)
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Ex 4. Find the intervals of increase/decrease
and max/min points of f(x) = x2e-x
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Ex 5. Find the local and absolute maximum
& minimum points for f(x) = x(ln(x))2
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Ex 6. For the function f(x) = xex determine
the co-ordinates of and classify the
extrema using the SDT
Ex 7. For the function f(x) = 2sin(x) +
sin2(x) determine the co-ordinates of and
classify the extrema using the SDT
Ex 8. Determine the co-ordinates of and
classify the extrema using the SDT for
ln x
f (x) 
x
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Textbook, S5.1 (FDT), p278 - 282
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(i) Q7,8 (graphs)
(ii) Q9-26 (algebra, ANV)
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Textbook, S5.2 (SDT), p307-310,
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(i) Graphs: Q27-32
(ii) Algebra: max/min; Q33-44, ANV
(iii) Algebra: SDT; Q48-53, ANV
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Visual Calculus - Maxima and Minima from
UTK
Visual Calculus - Mean Value Theorem and
the First Derivative Test from UTK
First Derivative Test -- From MathWorld
Tutorial: Maxima and Minima from Stefan
Waner at Hofstra U
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Graphing Using First and Second Derivatives from UC Davis
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Visual Calculus - Graphs and Derivatives from UTK
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Calculus I (Math 2413) - Applications of Derivatives - The
Shape of a Graph, Part II Using the Second Derivative - from
Paul Dawkins
http://www.geocities.com/CapeCanaveral/Launchpad/2426/
page203.html
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