Average Velocity and Instantaneous Velocity

Trip from CC-San Antonio

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

What was the average velocity between the second and fourth hour?

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The average velocity is the average rate of change of the distance on a given time interval.

This idea will be used to discussed the concept of instantaneous velocity at any point.

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Average Velocity From the Right

How does the police determine your velocity after 3 seconds?

Geometrically, this number represents the slope of the line segment passing through the points (3,900) and (6,3600).

Slope of the secant line is

Distance traveled by a car with the gas pedal pressed down 7/8 of the way

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Complete the table below. First, use the graph of the distance function to estimate the average velocity on each of the intervals. Second, calculate those averages using the formula that defines the function.

Use the calculator to find the average velocity on the intervals [3, 3.1], [3,3.01], [3,3.001].

Estimate

EXERCISE 1

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Complete the table below. First, use the graph of the distance function above to estimate the average velocity on each of the intervals. Second, calculate those averages using the formula that defines the function.

Use the calculator to find the average velocity on the intervals [2.9, 3], [2.95, 3], [2.98, 3].

EXERCISE 2

Estimate

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Right Instantaneous VelocityInstantaneous velocity at t=3 from the right,

Any point to the right of 3 can be written as

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Left Instantaneous Velocity

Instantaneous velocity at t=3 from the left

Any point to the left of 3 can be written as

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Since it is said that the instantaneous velocity at t=3 is 600.

Instantaneous velocity (or simply velocity) at t=3 is the instantaneous rate of change of the distance function at t=3.

Instantaneous Velocity at t=3

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Instantaneous Velocity at t=3

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Geometric Interpretation of the Instantaneous Velocity at t=3

The instantaneous velocity at t=3 is the slope of the tangent line to the function d=d(t) at t=3.

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Equation Of The Tangent Line at t = 3 Using The Instantaneous Velocity

Point: (3,900)Slope: v(3)=600

Equation of tangent line at the point (3,900), or when t=3 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=3.

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=3, when the interval in the domain containing 3 "shrinks".

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Key Relationship Between the Distance Function and The Tangent Line

Nearby t=3, the values of are about the same. It is, for values “close to t=3”

Nearby the point (3,900) the graph of d=d(t) and its tangent

line “look alike”