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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


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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


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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


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Average Change

  • Ex: r(t) = pool sales


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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


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Average Change

Rate of change from April to August

r(8) - r(4)


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Average Change

Average rate of change from April to August

r(8) - r(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

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


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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

Average rate of change from April to August

rise

run


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Average Change

Average change between two points

Slope of the secant line between the two points

=


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Instantaneous Rate of Change

  • Rate of change at this instant

    • Average rate of change over an infinitesimally small range


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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 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 don’t intersect graph at the point of tangency, but

  • Tangent lines can intersect graph at other points


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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

  • Concave Down


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Instantaneous Rates of Change

  • Concave Up


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Instantaneous Rates of Change

  • Exists only where you have a continuous function

  • Does not exist at breakpoints


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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


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Instantaneous Rates of Change

  • In-Class

    • Pg 185: 7, 8, 9, 10


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Derivatives

Section 3.3

Section 4.1


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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


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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


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Derivatives

  • Notation

“Derivative of f with respect to x”


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Derivatives

  • Ex: Profit versus number of employees

  • p(t) = 20*ln(t)


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Derivatives


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Derivatives

  • Notation

  • Interpretation


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Derivatives

  • In-Class

    • Pg 203: 1, 2, 4, 6, 10


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