Average Velocity and Instantaneous Velocity

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In a five-hour trip you traveled 300 miles. What was the average velocity for the whole trip?

The average velocity is nothing else that the average rate of change of the distance with respect to time.

This idea will be used to discussed the idea of instantaneous velocity as well as the instantaneous

rate of change of any continuous function

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Sided Average velocityHow does the police determine your velocity after 2 seconds?

Geometrically, this number represents the slope of the line segment passing through the points (2,400) and (4,1600).

Slope of the secant line is

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Exercise 1Estimate the average velocity on each of the intervals below. Include the units. Sketch on the graph the lines used to estimate those average velocities and estimate their slopes. Now, calculate those averages using the formula that defines the function.

Conjecture what is the average velocity when the interval is [2,b], b greater than 2 but "very" close to 2.

Estimate

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Exercise 2Estimate the average velocity on each of the intervals below. Include the units. Sketch on the graph the lines used to estimate those average velocities and estimate their slopes. Now, calculate those averages using the formula that defines the function.

Conjecture what is the average velocity when the interval is [b,2], b less than 2 but "very" close to 2.

Estimate

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Left Instantaneous velocity

Instantaneous velocity at t=2 from the left

Any point to the left of 2 can be written as

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Right Instantaneous velocity

Instantaneous velocity at t=2 from the right,

Any point to the right of 2 can be written as

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Since it is said that the instantaneous velocity at t=2 is 400.

Instantaneous velocity (or simply velocity) at t=2 is the instantaneous rate of change of the distance function at t=2.

Instantaneous Velocity at t=2

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Instantaneous Velocity at t=2

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Geometric Interpretation of the Instantaneous Velocity at t=2

The instantaneous velocity at t=2, v(2) is the slope of the tangent line to the function d=d(t) at

t=2.

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Equation Of The Tangent Line at t = 2 Using The Instantaneous Velocity

Point: (2,400)Slope: v(2)=400

Equation of tangent line at the point (2,400), or when t=2 is

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COMPARING THE GRAPH OF THE FUNCTION AND THE TANGENT LINE

NEARBY THE TANGENCY POINT

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The graphs of and are displayed on windows where the domain are intervals "shrinking" around t=2.

a. In your graphing calculator reproduce the graphs above. Make sure you in each case the windows have the same dimensions.

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In a paragraph, and in your own language, explain what happens to the graphs of the distance function and its tangent line at t=2, when the interval in the domain containing 2 "shrinks".

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Important Relationship

Nearby t=2, the values of are about the same. It is, nearby t=2. Or