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## PowerPoint Slideshow about 'Instantaneous Rate of Change' - winka

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Average Change

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

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

Average Change

- Ex: r(t) = pool sales

Average Change

- Rate of change
- Percent change
- Average change

Average Change

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

Instantaneous Rate of Change

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

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

Instantaneous Rates of Change

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

Instantaneous Rates of Change

- Tangent lines don’t intersect graph at the point of tangency, but
- Tangent lines can intersect graph at other points

Instantaneous Rates of Change

- Concave Down

Instantaneous Rates of Change

- Concave Up

Instantaneous Rates of Change

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

Instantaneous Rates of Change

- In-Class
- Pg 185: 7, 8, 9, 10

Derivatives

- Another phrase for instantaneous rate of change

Instantaneous rate of change

Rate of change

=

=

Slope of curve

Slope of tangent line

=

=

Derivative

Derivatives

- Ex: Profit versus number of employees
- p(t) = 20*ln(t)

Derivatives

- Notation
- Interpretation

Derivatives

- In-Class
- Pg 203: 1, 2, 4, 6, 10

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