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Instantaneous Rate of Change

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Instantaneous Rate of Change

Sections 3.1-3.3

Section 4.1

- Defined equations in terms of their changes
- e.g., exponential constant percentage change

- Will use this concept motivate derivatives

- 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

- Ex: r(t) = pool sales

- Rate of change
- Percent change
- Average change

Rate of change from April to August

Rate of change

Rate of change from April to August

r(8) - r(4)

Average rate of change from April to August

r(8) - r(4)

Average rate of change from April to August

r(8) - r(4)

8-4

Average rate of change from April to August

r(8) - r(4)

8-4

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

Average rate of change from April to August

r(8) - r(4)

8-4

Average rate of change from April to August

rise

run

Average change between two points

Slope of the secant line between the two points

=

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

- 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

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

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

- Concave Down

- Concave Up

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

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

Derivatives

Section 3.3

Section 4.1

- Another phrase for instantaneous rate of change

Instantaneous rate of change

Rate of change

=

=

Slope of curve

Slope of tangent line

=

=

Derivative

- Notation

“Derivative of f with respect to x”

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

- Notation
- Interpretation

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