We can define the instantanous velocity as a limit of an average velocity,
as the time interval gets smaller and smaller. Let *s* (*t*) be the
position of an object at time *t*. The instantaneous velocity at *t* = *a* is defined as
.

See About the calculus applets for operating instructions. |

Initially, the applet shows a graph of height above the ground versus time, for
the object that we examined in the previous page. Note that this graph is
**not** a drawing of the path of the object, but is a graph
of height versus time. The object was actually tossed straight up and
fell straight back down.

Select the second example from the drop down menu. This just zooms in on the interval between 0 and 3 seconds. Notice the green line, which extends from the green dot at the point (2,142) to the red dot at the point (1,90). What is the slope of this green line? It's just rise over run, or But this is also the average velocity over the interval from 1 to 2 seconds. In other words, we can visualize the average velocity over an interval as the slope of the secant line between the endpoints of that interval. The slope of the green secant line is displayed in a small box on the graph.

What we want to find out is the instantaneous velocity at *t* = 1
second. We can approach this just like on the previous page by making the
interval smaller. Click-drag the green dot closer to the red dot (zooming
with the mouse is turned off on this applet so you don't accidentally
zoom when you really meant to click on a dot). Notice that the slope is
increasing as the green dot approaches the red dot.

In fact, the slope of
the secant is approaching the slope of the red line, which is tangent to
the curve at the point (1,90). The slope of this tangent line is 68,
which is the instantaneous velocity at *t* = 1. We can think of
instantaneous velocity as the slope of the tangent line at a point on our
position curve, just like average velocity is the slope of the secant
line.

- 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

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved