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Instantaneous Rate of Change. Sections 3.1-3.3 Section 4.1. 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

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Instantaneous Rate of Change

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Instantaneous Rate of Change

Sections 3.1-3.3

Section 4.1


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


Average Change

  • Rate of change

  • Percent change

  • Average change


Average Change

Rate of change from April to August

Rate of change


Average Change

Rate of change from April to August

r(8) - r(4)


Average Change

Average rate of change from April to August

r(8) - r(4)


Average Change

Average rate of change from April to August

r(8) - r(4)

8-4


Average Change

Average rate of change from April to August

r(8) - r(4)

8-4


Average Change

  • In-Class

    • Pg 167: 1, 2, 4, 6, 7, 8, 10


Average Change

Average rate of change from April to August

r(8) - r(4)

8-4


Average Change

Average rate of change from April to August

rise

run


Average Change

Average change between two points

Slope of the secant line between the two points

=


Instantaneous Rate of Change

  • Rate of change at this instant

    • Average rate of change over an infinitesimally small range


Instantaneous Rate of Change


Instantaneous Rate of Change


Instantaneous Rate of Change


Instantaneous Rate of Change


Instantaneous Rate of Change


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 Rate of Change


Instantaneous Rate of Change


Instantaneous Rate of Change


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


Instantaneous Rates of Change


Instantaneous Rates of Change


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


Instantaneous Rates of Change


Instantaneous Rates of Change


Instantaneous Rates of Change


Instantaneous Rates of Change

  • In-Class

    • Pg 185: 7, 8, 9, 10


Derivatives

Section 3.3

Section 4.1


Derivatives


Derivatives


Derivatives


Derivatives


Derivatives


Derivatives

  • Another phrase for instantaneous rate of change

Instantaneous rate of change

Rate of change

=

=

Slope of curve

Slope of tangent line

=

=

Derivative


Derivatives

  • Notation

“Derivative of f with respect to x”


Derivatives

  • Ex: Profit versus number of employees

  • p(t) = 20*ln(t)


Derivatives


Derivatives

  • Notation

  • Interpretation


Derivatives

  • In-Class

    • Pg 203: 1, 2, 4, 6, 10


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