MAT 166 - Chapter 3 Slides - Chapter 3 Slides.pdf · 5/29/2013 10 Continuity (Poultry Farming, #41)...
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5/29/2013 1 MAT 166 – Calculus for Bus/Soc Chapter 3 Notes The Deriviative David J. Gisch Limits Limits Limits
Transcript of MAT 166 - Chapter 3 Slides - Chapter 3 Slides.pdf · 5/29/2013 10 Continuity (Poultry Farming, #41)...
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MAT 166 – Calculus for Bus/SocChapter 3 NotesThe DeriviativeDavid J. Gisch
Limits
Limits Limits
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Limits• Why? Don’t they always agree?
Limits
22
LimitsExample: What is the limit of
21 2
as → 2?
Limits
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Limits LimitsExample: What is the limit, if it exists of
lim→
Rules for Limits Rules for Limits
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Algebra of LimitsExample: Suppose that
lim→
3 lim→
4
What is lim→
?
Algebra of LimitsExample: Find the limit, if it exists, for
lim→
123
Algebra of LimitsExample: Find the limit, if it exists, for
lim→
11
Algebra of LimitsExample: Find the limit, if it exists, for
lim→
24
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Limits at Infinity• Sometimes it is very useful to look at the “end” behavior
of a graph. We do this by contemplating the limit at infinity or negative infinity.
Example: Suppose a small pond normally contains 12 units of dissolved oxygen in a fixed volume of water. Suppose also that at time t=0 a quantity of organic waste is introduced into the pond, with the oxygen concentration t weeks later given by
12 15 121
As time goes on, what will be the ultimate concentration of oxygen?
Limits at Infinity (Graphically)
lim→
12 15 121
Algebra of LimitsExample: Find the limit, if it exists, for
lim→
2 46 5 7
Solution: Here, the highest power of (in the denominator) is , which is used to divide each term in the numerator and denominator.
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Continuity
Continuity
ContinuityExample: Examine the graphs and decide if they are continuous at the indicated point.
At 3 At 0
ContinuityExample: Examine the graphs and decide if they are continuous at the indicated point.
At 4
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ContinuityExample: Examine the graphs and decide if they are continuous at the indicated point.
At 2
ContinuityExample: Examine the graphs and decide if they are continuous at the indicated point.
At 1
Continuity on Closed Intervals Continuity of Functions
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Continuity of Functions Shifts in Relation to Continuity• Understand that the last charts refer to “parent” graphs.• For example
• But what if I have the function log 5 .
ContinuityExample: Find all values where the function is discontinuous.
4 32 7
ContinuityExample: Find all values where the function is discontinuous.
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ContinuityExample: Find all values where the function is discontinuous.
1 13 4 1 3
5 3
ContinuityExample: Find all values where the function is discontinuous.
27 7 2 8
a) 1,
b)
c) 0, 1,
d) 1,
Continuity (Poultry Farming, #41)Researchers at Iowa State University and the University o Arkansas have developed a piecewise function that can be used to estimate the body weight (in grams) of a male broiler during the first 56 days of life according to
48 3.64 0.6363 , 1 28 1004 65.8 , 28 56
Where t is the age of the chicken (in days).
What is the weight of a male broiler that is 25 days old?
Continuity (Poultry Farming, #41)48 3.64 0.6363 0.00963 , 1 28 1004 65.8 , 28 56
Is a continuous function?
Use a graphing calculator to graph on 1, 56 by 0, 3000 .
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Continuity (Poultry Farming, #41)48 3.64 0.6363 0.00963 , 1 28 1004 65.8 , 28 56
Why would researchers use two different types of functions to estimate the weight of a chicken at various ages?
Rates of Change
Average Rate of Change
• Average rate of change looks a lit like slope and it is in some sense. If a function is linear they are the same thing. If a function is not linear then they are two different things.
Rate of ChangeExample: Find the average rate of change for the function on the indicated intervals.
2 6 4
a) 0, 6
b) 4, 10
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Average Rate of Change Average Rate of Change
Average Rate of Change Average Rate of Change• Understand that Average Rate of Change is exactly what
it says, an average.• To see what we mean lets look at the next example.
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Average Rate of ChangeExample: Find the average rate of change for each function on the 0, 10 .
5 15 2 5
Instantaneous Rate of Change
Recall that .
Instantaneous ROC• What is going on visually?
Instant ROCExample: Find the instantaneous rate of change for the function 2 6 4 at the point 4.
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Practical Use• We can often create functions describing the position of
an object.• How does you position change?▫ Changes based on velocity (speed).
Book Example: P. 10 #37
Definition of Derivative
Tangent Line
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Tangent Line (Instant ROC) Tangent Line
Tangent LineExample: Consider the graph of the function 2.
a) Find the slope and equation of the secant line through the points where 1 and 2.
Tangent LineExample: Consider the graph of the function 2.
b) Find the slope and equation of the tangent line at 1.
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Graph of Example Genetics ExampleThe graph below shows the risk of chromosomal abnormalities in a child increases with the age of the mother at the child's birth. Find the rate that the risk is rising when the mother is 40 years old.
The Derivative The DerivativeExample: Consider the function .
a) Find the derivative of (i.e. find ′ .
b) Find ′ 5 .
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The DerivativeExample: Consider the function 2 4 .
a) Find the derivative of (i.e. find ′ .
b) Find ′ 3 .
Existence Existence
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Existence
Graphical Differentiation
Graphical Differentiation• It is helpful to be able to look at a graph and be able
estimate what the graph of its derivative looks like.
• For each of the following graphs sketch the graph of its derivative.
Graphical Differentiation
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Graphical Differentiation
