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Two Questions of Calculus

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Objective To determine the purpose for studying calculus. TS: Making decisions after reflection and review

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Two Questions of Calculus I went on a bike ride.

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Two Questions of Calculus Two-thirds of the way, I saw a sign.

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Two Questions of Calculus Did I break the law?

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Two Questions of Calculus Did I break the law, the instant I passed the sign?

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First Question of Calculus How do you find instantaneous rates of change (velocities)?

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Distance, Rate, & Time These are very useful for finding average rates, or for finding total distance from average rates, etc.

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Distance, Rate, and Time D R T

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Second Question of Calculus How do you find areas (or volumes) of exotic shapes?

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Two Questions of Calculus How do you find instantaneous rates of change (velocities)? How do you find areas (or volumes) of exotic shapes? These two questions have related answers.

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Rates of Change Velocity – the rate of change of position (Slope or Rate of change of distance v. time) Acceleration – the rate of change of velocity (Slope or Rate of change of velocity v. time) Population growth rates Unemployment rates Production rates

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Two Types of Rate of Change Average – Rate of change between 2 points Ex: average speed traveled during a trip The average rate of change = the slope of the line containing the two points (so this is a secant slope)

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Two Types of Rate of Change

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Instantaneous – Rate of change at one point Ex: speed traveled at a single moment during a trip

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Two Types of Rate of Change

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Chapter II: The Derivative A common rate of change that we will be using is average velocity.

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Chapter II: The Derivative Positive velocities are used for upward motion. Negative velocities used for downward motion. The absolute value of velocity is the speed of an object.

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