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.

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Presentation transcript:

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?

The average velocity is the average rate of change of the distance on a given time interval. 3 This idea will be used to discussed the concept of instantaneous velocity at any point.

4

Average Velocity From the Right How does the police determine your velocity after 3 seconds? 5 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

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]. 6 Estimate EXERCISE 1

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]. 7 EXERCISE 2 Estimate

Right Instantaneous Velocity Instantaneous velocity at t=3 from the right, 8 Any point to the right of 3 can be written as

Left Instantaneous Velocity Instantaneous velocity at t=3 from the left 9 Any point to the left of 3 can be written as

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. 10 Instantaneous Velocity at t=3

11

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. 12

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 13

COMPARING THE GRAPH OF THE FUNCTION AND THE TANGENT LINE NEARBY THE TANGENCY POINT 14

The graphs of and are displayed on windows where the domain are intervals "shrinking" around t=3. 15 a. In your graphing calculator reproduce the graphs above. Make sure you in each case the windows have the same dimensions.

16 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".

Key Relationship Between the Distance Function and The Tangent Line 17 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”