The Tangent Problem and Differential Calculus Cont.
Calculating Instantaneous Velocity
Let’s continue our investigation by exploring a related application, instantaneous velocity. Velocity may be thought of as the rate of change of position.
Suppose that we have a function, [latex]s(t)[/latex], that gives the position of an object along a coordinate axis at any given time [latex]t[/latex]. Can we use these same ideas to create a reasonable definition of the instantaneous velocity at a given time [latex]t=a[/latex]?
We start by approximating the instantaneous velocity with an average velocity.
Recall that the speed of an object traveling at a constant rate is the ratio of the distance traveled to the length of time it has traveled.
We define the average velocity of an object over a time period to be the change in its position divided by the length of the time period.
average velocity
Let [latex]s(t)[/latex] be the position of an object moving along a coordinate axis at time [latex]t[/latex]. The average velocity of the object over a time interval [latex][a,t][/latex] where [latex]a < t[/latex] (or [latex][t,a][/latex] if [latex]t < a[/latex]) is
As [latex]t[/latex] is chosen closer to [latex]a[/latex], the average velocity becomes closer to the instantaneous velocity.
Note that finding the average velocity of a position function over a time interval is essentially the same as finding the slope of a secant line to a function. Furthermore, to find the slope of a tangent line at a point [latex]a[/latex], we let the [latex]x[/latex]-values approach [latex]a[/latex] in the slope of the secant line.
To find the instantaneous velocity at time [latex]a[/latex], we let the [latex]t[/latex]-values approach [latex]a[/latex] in the average velocity. This process of letting [latex]x[/latex] or [latex]t[/latex] approach [latex]a[/latex] in an expression is called taking a limit.
instantaneous velocity
For a position function [latex]s(t)[/latex], the instantaneous velocity at a time [latex]t=a[/latex] is the value that the average velocities approach on intervals of the form [latex][a,t][/latex] and [latex][t,a][/latex] as the values of [latex]t[/latex] become closer to [latex]a[/latex], provided such a value exists.
A rock is dropped from a height of [latex]64[/latex] ft. It is determined that its height (in feet) above ground [latex]t[/latex] seconds later (for [latex]0\le t\le 2[/latex]) is given by [latex]s(t)=-16t^2+64[/latex].
Find the average velocity of the rock over each of the given time intervals. Use this information to guess the instantaneous velocity of the rock at time [latex]t=0.5[/latex].
- [latex][0.49,0.5][/latex]
- [latex][0.5,0.51][/latex]