Let's take a look at average velocity. If you recall from earlier
mathematics studies, *average* *velocity* is just net distance
traveled divided by time. For example, if an object is tossed into the air we
might find the following data for the height in feet, *y*, of the object
as a function of the time in seconds, *t*, where *t* = 0 is when
the object is released upward.

t (sec) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

y (feet) | 6 | 90 | 142 | 162 | 150 | 106 | 30 |

This table says that the object was 6 feet above the ground when released,
90 feet after 1 second, etc. How fast was it traveling? We can divide the net
distanced traveled by the time to compute the velocity. For example, over the
first second the average velocity is
Over the last second, the
average velocity is
Note that velocity can be
negative to indicate downward motion (positive velocity is upward motion in
this problem). *Speed* is just the magnitude of the velocity (i.e.,
the absolute value, in our one-dimensional example). We can also compute the
average velocity over the entire interval as
Now, what if we wanted to know the velocity at a specific time, say at *t* = 1? Clearly there is some velocity, as the object is moving, but to
compute velocity we need a ratio of distance traveled to time spent
traveling. We could find the average velocity over the interval from 1 to 2
seconds, which is just
But this isn't quite right, as
the object does seem to be speeding up. If we had another measurement of
height at a time closer to 1 than 2 seconds, we could get a better estimate
of the instantaneous velocity at *t* = 1. The following applet
illustrates this.

See About the calculus applets for operating instructions. |

- The applet shows a subset of the table from above. At the bottom is a
value,
*h*, with an input box and a slider to control it. The first row of the table show the time at 1 second and the time at 1 +*h*seconds. The second row shows the height at these two times, while the third row computes the average velocity over this interval. Note that the initial view of the applet, with*h*= 1, just shows the average velocity between 1 and 2 seconds, as we computed above. - Move the slider to make
*h*smaller. You are essentially picking a smaller interval, so the average velocity should be a better approximation to the instantaneous velocity at*t*= 1. Keep making*h*smaller and smaller. What is happening to the velocity? Is it approaching some number?

- Make the velocity 0 by moving the slider all the way to the left. What happens to the velocity? Why?
- Type in 0.001 for
*h*and press Enter. Then try 0.0001, etc. What number is the velocity approaching as*h*approaches 0? - Note that if you set
*h*less than or equal to 0.000000001, round off errors start causing problems

- Constant, Line, and Power Functions
- Exponential Functions
- Trigonometric Functions
- Constant Multiple
- Combinations: Sum and Difference
- Combinations: Product and Quotient
- Composition of Functions (the Chain Rule)
- Transformations of Functions
- Inverses of Functions
- Hyperbolic Functions
- Linear Approximation
- Mean Value Theorem

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