Chapter 4.1.notebook
October 17, 2016
Chapter 4 - Applications of the Derivative
4.1 Maxima and Minima
4.2 What Derivatives Tell Us
4.3 Graphing Functions
4.4 Optimization Problems
4.5 Linear Approximation and Differentials
4.6 Mean Value Theorem
4.7 L'Hopital's Rule
4.8 Newton's Method
4.9 Antiderivatives
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Vocabulary:
1.
2.
3.
4.
Absolute Maximum and Local Minimum Values
Local Maximum and Minimum Values
Extreme Values - Local or Absolute
Critical Points
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Definition: Absolute Maximum and Minimum Values
Let f(x) be defined on an interval I containing x = c.
a) If f(c) ≥ f(x) for every x in I, then
f(x) has an absolute maximum value of f(c) on I at x = c.
b) If f(c) ≤ f(x) for every x in I, then
f(x) has an absolute minimum value of f(c) on I at x = c.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
How to find Absolute Values for different interval types: closed, open,
half-open/closed.
a) The graph can be used initially.
b) We solve for it....stay tuned!
Let's look at graphs first:
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
How to find Absolute Values for different interval types: closed, open,
half-open/closed.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Theorem 4.1 Extreme Value Theorem
A function that is continuous on a closed interval [a,b] has an absolute maximum value,
f(c) AND an absolute minimum value f(d) where x = d and x = c are in interval [a,b].
Exercise: Pick any closed interval then pick
other interval types to see that the theorem
doesn't guarantee an absolute max or min
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Theorem 4.1 Extreme Value Theorem
A function that is continuous on a closed interval [a,b] has an absolute maximum value,
f(c) AND an absolute minimum value f(d) where x = d and x = c are in interval [a,b].
Note: [a,b] is closed but if f(x) is not continuous in [a,b]
then no absolute max or min is guaranteed.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Definition: Local Maximum and Minimum Values
Suppose f(x) is defined on any interval I and x=c is an interior point of the interval.
a) If f(c) ≥ f(x) for all x 'near x=c' then f(c) is a local maximum value of f(x).
b) If f(c) ≤ f(x) for all x 'near x=c' then f(c) is a local minimum value of f(x).
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Definition: Local Maximum and Minimum Values
Suppose f(x) is defined on any interval I and x=c is an interior point of the interval.
a) If f(c) ≥ f(x) for all x 'near x=c' then f(c) is a local maximum value of f(x).
b) If f(c) ≤ f(x) for all x 'near x=c' then f(c) is a local minimum value of f(x).
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
How do we know if f(x) has an local max or min on an interval?
a) The graph can be used initially.
b) We solve for it using theorem 4.2, the Local Extreme Point Theorem
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Definition: Critical Point.
An interior point x = c of the domain of f(x) for which f '(c) = 0 or f ' (c) does not exist is
called a critical point.
Theorem 4.2 Local Extreme Point theorem....This is how we solve for POSSIBLE local
maximum and minimum values.
If f(x) has a local maximum or minimum value at x = c and if f '(x) exists then f '(c) = 0
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Locate the critical points of the function. These are possible local
max and/or min of the function.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Locate the critical points of the function. These are possible local
max and/or min of the function.
over [-4, 4]
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Locate the critical points of the function. These are possible local
max and/or min of the function.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Locate the critical points of the function. These are possible local
max and/or min of the function.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Locate the critical points of the function. These are possible local
max and/or min of the function.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Procedure: Finding Absolute Maximum and Minimum Values (extreme).
Assume the function f(x) is continuous on the closed interval [a,b]
1. Find the critical points. These are possible extreme values.
2. Evaluate f(x) at the critical points and the endpoints of [a,b]
3. Choose the largest value of f(a), f(b), f(c) for the absolute max. and
the smallest value of f(a), f(b), f(c) for the absolute min.
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Find the Absolute Maximum and Minimum Values (extreme).
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Find the Absolute Maximum and Minimum Values (extreme).
Suppose a tour guide has a bus that holds a maximum of 100 people. Assume
his profit (in dollars) for taking 'n' people on a city tour is
P(n) = n (50 - 0.5n) - 100.
a) How many people should the guide take on a tour to maximize profit?
b) Suppose the bus holds a maximum of 45 people. How many people
should
be taken on a tour to maximize the profit?
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Chapter 4.1.notebook
October 17, 2016
Chapter 4.1 - Maxima and Minima
Exercise: Find the Absolute Maximum and Minimum Values (extreme).
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