MAT 166 Calculus for Bus/Soc Chapter 3 Notes Limits The - - PowerPoint PPT Presentation

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MAT 166 – Calculus for Bus/Soc Chapter 3 Notes

The Deriviative David J. Gisch

Limits

Limits Limits

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Limits

• Why? Don’t they always agree?

Limits

2 2

Limits

Example: What is the limit of

2 1 2

as → 2?

Limits

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Limits Limits

Example: What is the limit, if it exists of

lim

→

• Rules for Limits

Rules for Limits

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Algebra of Limits

Example: Suppose that

lim

→ 3 lim → 4

What is lim

→ ?

Algebra of Limits

Example: Find the limit, if it exists, for

lim

→

12 3

Algebra of Limits

Example: Find the limit, if it exists, for

lim

→

1 1

Algebra of Limits

Example: Find the limit, if it exists, for

lim

→

2 4

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Limits at Infinity

• Sometimes it is very useful to look at the “end” behavior
• f a graph. We do this by contemplating the limit at

infinity or negative infinity. Example: Suppose a small pond normally contains 12 units

• f 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 12 1 As time goes on, what will be the ultimate concentration of

• xygen?

Limits at Infinity (Graphically)

lim

→

12 15 12 1

Algebra of Limits

Example: Find the limit, if it exists, for

lim

→

2 4 6 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 Continuity

Example: Examine the graphs and decide if they are continuous at the indicated point.

At 3 At 0

Continuity

Example: Examine the graphs and decide if they are continuous at the indicated point.

At 4

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Continuity

Example: Examine the graphs and decide if they are continuous at the indicated point.

At 2

Continuity

Example: 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 S hifts in Relation to Continuity

• Understand that the last charts refer to “parent” graphs.
• For example
• But what if I have the function log 5.

Continuity

Example: Find all values where the function is discontinuous. 4 3 2 7

Continuity

Example: Find all values where the function is discontinuous.

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Continuity

Example: Find all values where the function is discontinuous. 1 1 3 4 1 3 5 3

Continuity

Example: Find all values where the function is discontinuous. 2 7 72 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 Change

Example: Find the average rate of change for the function

• n 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 Change

Example: Find the average rate of change for each function

• n the 0, 10 .

5 1 5 2 5

Instantaneous Rate of Change

Recall that .

Instantaneous ROC

• What is going on visually?

Instant ROC

Example: 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 Line

Example: 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 Line

Example: 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 Example

The 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 Derivative

Example: Consider the function . a) Find the derivative of (i.e. find ′. b) Find ′5.

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The Derivative

Example: 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