Instantaneous Rate of Change

1
Instantaneous Rate of Change

• Sections 3.1-3.3
• Section 4.1

2
Average Change

• Defined equations in terms of their changes
• e.g., exponential ? constant percentage change
• Will use this concept motivate derivatives

3
Average Change

• Rate of change
• Difference between two values
• Percentage change
• Difference between two values as a percentage of
  the original value
• Average change
• Change per unit of time

4
Average Change

• Ex r(t) pool sales

Month Sales
1 2
2 3
3 6
4 10
5 15
6 21
7 18
8 15
9 9
10 7
11 3
12 1
5
Average Change
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Average Change

• Rate of change
• Percent change
• Average change

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Average Change
Rate of change from April to August
Rate of change
8
Average Change
Rate of change from April to August
r(8) - r(4)
9
Average Change
Average rate of change from April to August
r(8) - r(4)
10
Average Change
Average rate of change from April to August
r(8) - r(4)
8-4
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Average Change
Average rate of change from April to August
r(8) - r(4)
8-4
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Average Change

• In-Class
• Pg 167 1, 2, 4, 6, 7, 8, 10

13
Average Change
Average rate of change from April to August
r(8) - r(4)
8-4
14
Average Change
Average rate of change from April to August
rise
run
15
Average Change
Average change between two points
Slope of the secant line between the two points

16
Instantaneous Rate of Change

• Rate of change at this instant
• Average rate of change over an infinitesimally
  small range

17
Instantaneous Rate of Change
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Instantaneous Rate of Change
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Instantaneous Rate of Change
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Instantaneous Rate of Change
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Instantaneous Rate of Change
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Instantaneous Rates of Change

• Tangent line
• Secant line that touches the graph at the point
  evaluated

The instantaneous rate of change is the slope of
the tangent line at the point evaluated
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Instantaneous Rates of Change

• Local linearity
• Zoom in enough and anything looks like a line.

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Instantaneous Rate of Change
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Instantaneous Rate of Change
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Instantaneous Rate of Change
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Instantaneous Rates of Change

• Tangent lines dont intersect graph at the point
  of tangency, but
• Tangent lines can intersect graph at other points

28
Instantaneous Rates of Change
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Instantaneous Rates of Change
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Instantaneous Rates of Change
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Instantaneous Rates of Change

• Concave Down

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Instantaneous Rates of Change

• Concave Up

33
Instantaneous Rates of Change

• Exists only where you have a continuous function
• Does not exist at breakpoints

34
Instantaneous Rates of Change
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Instantaneous Rates of Change
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Instantaneous Rates of Change
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Instantaneous Rates of Change
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Instantaneous Rates of Change

• In-Class
• Pg 185 7, 8, 9, 10

39
Derivatives

• Section 3.3
• Section 4.1

40
Derivatives
Year Value
0 3
1 3.1
2 1.8
3 -0.3
4 -2.6
5 -4.5
6 -5.4
7 -4.7
8 -1.8
9 3.9
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Derivatives
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Derivatives
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Derivatives
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Derivatives
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Derivatives

• Another phrase for instantaneous rate of change

Instantaneous rate of change
Rate of change

Slope of curve

Slope of tangent line

Derivative

46
Derivatives

• Notation

Derivative of f with respect to x
47
Derivatives

• Ex Profit versus number of employees
• p(t) 20ln(t)

48
Derivatives
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Derivatives

• Notation
• Interpretation

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Derivatives

• In-Class
• Pg 203 1, 2, 4, 6, 10