Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4

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Chapter 5Applications of the DerivativeSections
5.1, 5.2, 5.3, and 5.4
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Applications of the Derivative

• Maxima and Minima
• Applications of Maxima and Minima
• The Second Derivative - Analyzing Graphs

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Absolute Extrema
Let f be a function defined on a domain D
Absolute Maximum
Absolute Minimum
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Absolute Extrema
A function f has an absolute (global) maximum at
x c if f (x) ? f (c) for all x in the domain D
of f.
The number f (c) is called the absolute maximum
value of f in D
Absolute Maximum
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Absolute Extrema
A function f has an absolute (global) minimum at
x c if f (c) ? f (x) for all x in the domain D
of f.
The number f (c) is called the absolute minimum
value of f in D
Absolute Minimum
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Generic Example
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Generic Example
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Generic Example
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Relative Extrema
A function f has a relative (local) maximum at x
? c if there exists an open interval (r, s)
containing c such that f (x) ? f (c) for all r
? x ? s.
Relative Maxima
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Relative Extrema
A function f has a relative (local) minimum at x
? c if there exists an open interval (r, s)
containing c such that f (c) ? f (x) for all r
? x ? s.
Relative Minima
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Generic Example
The corresponding values of x are called
Critical Points of f
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Critical Points of f
A critical number of a function f is a number c
in the domain of f such that
(stationary point)
(singular point)
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Candidates for Relative Extrema

1. Stationary points any x such that x is in the
   domain of f and f ' (x) ? 0.
2. Singular points any x such that x is in the
   domain of f and f ' (x) ? undefined
3. Remark notice that not every critical number
   correspond to a local maximum or local minimum.
   We use local extrema to refer to either a max
   or a min.

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Fermats Theorem
If a function f has a local maximum or minimum
at c, then c is a critical number of f
Notice that the theorem does not say that at
every critical number the function has a local
maximum or local minimum
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Generic Example
Two critical points of f that do not correspond
to local extrema
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Example
Find all the critical numbers of
Stationary points
Singular points
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Graph of
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Extreme Value Theorem
If a function f is continuous on a closed
interval a, b, then f attains an absolute
maximum and absolute minimum on a, b. Each
extremum occurs at a critical number or at an
endpoint.
a b
a b
a b
Attains max. and min.
Attains min. but no max.
No min. and no max.
Open Interval
Not continuous
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Finding absolute extrema on a , b

1. Find all critical numbers for f (x) in (a , b).
2. Evaluate f (x) for all critical numbers in (a ,
   b).
3. Evaluate f (x) for the endpoints a and b of the
   interval a , b.
4. The largest value found in steps 2 and 3 is the
   absolute maximum for f on the interval a , b,
   and the smallest value found is the absolute
   minimum for f on a , b.

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Example
Find the absolute extrema of
Critical values of f inside the interval (-1/2,3)
are x 0, 2
Evaluate
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Example
Find the absolute extrema of
Critical values of f inside the interval (-1/2,3)
are x 0, 2
Absolute Max.
Absolute Min.
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Example
Find the absolute extrema of
Critical values of f inside the interval (-1/2,1)
is x 0 only
Absolute Max.
Evaluate
Absolute Min.
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Example
Find the absolute extrema of
Critical values of f inside the interval (-1/2,1)
is x 0 only
Absolute Max.
Absolute Min.
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Increasing/Decreasing/Constant
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Increasing/Decreasing/Constant
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Increasing/Decreasing/Constant
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The First Derivative Test
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Generic Example
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The First Derivative Test
Determine the sign of the derivative of f to
the left and right of the critical point.
left
right
conclusion
f (c) is a relative maximum
f (c) is a relative minimum
No change
No relative extremum
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Relative Extrema
Example Find all the relative extrema of
Stationary points
Singular points None
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The First Derivative Test
Find all the relative extrema of
0 - 0
0 4
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The First Derivative Test
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The First Derivative Test
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Another Example
Find all the relative extrema of
Stationary points
Singular points
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Stationary points
Singular points
Relative max.
Relative min.
ND 0 - ND - 0
ND
-1 0 1
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Graph of
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Domain Not a Closed Interval
Example Find the absolute extrema of
Notice that the interval is not closed. Look
graphically
Absolute Max.
(3, 1)
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Optimization Problems

1. Identify the unknown(s). Draw and label a
   diagram as needed.

1. Identify the objective function. The quantity
   to be minimized or maximized.

1. Identify the constraints.

4. State the optimization problem.
5. Eliminate extra variables.
6. Find the absolute maximum (minimum) of the
objective function.
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Optimization - Examples
An open box is formed by cutting identical
squares from each corner of a 4 in. by 4 in.
sheet of paper. Find the dimensions of the box
that will yield the maximum volume.
x
4 2x
x
x
x
4 2x
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Critical points
The dimensions are 8/3 in. by 8/3 in. by 2/3 in.
giving a maximum box volume of V ? 4.74 in3.
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Optimization - Examples
An metal can with volume 60 in3 is to be
constructed in the shape of a right circular
cylinder. If the cost of the material for the
side is 0.05/in.2 and the cost of the material
for the top and bottom is 0.03/in.2 Find the
dimensions of the can that will minimize the cost.
top and bottom
side
cost
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So
Sub. in for h
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Graph of cost function to verify absolute minimum
2.5
So with a radius 2.52 in. and height 3.02 in.
the cost is minimized at 3.58.
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Second Derivative
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Second Derivative - Example
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Second Derivative
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Second Derivative
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Concavity
Let f be a differentiable function on (a, b).
1. f is concave upward on (a, b) if f ' is
increasing on aa(a, b). That is f ''(x) ? 0 for
each value of x in (a, b).
2. f is concave downward on (a, b) if f ' is
decreasing on (a, b). That is f ''(x) ? 0 for
each value of x in (a, b).
concave upward
concave downward
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Inflection Point
A point on the graph of f at which f is
continuous and concavity changes is called an
inflection point.
To search for inflection points, find any point,
c in the domain where f ''(x) ? 0 or f ''(x) is
undefined. If f '' changes sign from the left to
the right of c, then (c, f (c)) is an
inflection point of f.
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Example Inflection Points
Find all inflection points of
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Inflection point at x ? 2
- 0
2
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The Point of Diminishing Returns
If the function
represents the total sales of a particular
object, t months after being introduced, find the
point of diminishing returns.
S concave up on
S concave down on
The point of diminishing returns is at 20 months
(the rate at which units are sold starts to drop).
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The Point of Diminishing Returns
S concave down on
Inflection point
S concave up on