2.2.5 Notation and Terminology

There is still more notation and terminology, even without a single additional concept. The instantaneous rate of change (think derivative) of position or distance is what we have been calling “instantaneous speed” — at least for an object that moves in only one direction along a straight line. For motion in general, this rate is called velocity. We abbreviate the sentence,

to

Finally, when we want to focus on functional notation, we use a notation for “the derived function”: If $s=f(t),$ then $v=f^{\prime }\left(t\right)$ (which is read, “f prime of t”). The notations $f^{\prime }$ and \(ds / dt\) are interchangeable. Each will be used when it is convenient for the task at hand. Thus, for Example 1,

and

The Graph pop-up below is designed for Activity 5 following. It will enable you to zoom in on the graph of a function of your choice, selecting the "zoom center" and the "zoom power". If the zoom center is \(t_1\) and the zoom power is \(p\), you will be plotting the function on the interval \( [t_1-\Delta t, t_1+\Delta t] \), where \(\Delta t=1/2^{p}\). That is, each increase of \(1\) in \(p\) shortens the interval by a factor of \(1/2\). When your plot looks straight, the grid lines will help you find coordinates for a slope calculation.

Activity 5

Suppose $s=2t^2+3.$

1. 

   Use our graphing tool or your own graphing calculator to determine the values of \(ds/dt\) at \(t = 1\), $t=\mathrm{2,}$ and $t=3.$ Our pop-up calculator can help with the numeric calculations.

2. Conjecture a formula for \(ds / dt\) that agrees with your calculation in part (a).

3. Use algebra to calculate \(\Delta s / \Delta t\) where $\Delta t=t_2-t_1$.

4. From your calculation in part (c), write down a formula for \(ds / dt\) at $t_1$.

5. Write down a formula for \(ds / dt\) as a function of \(t\). [This requires only a simple modification of your formula in part (d).] Does your formula agree with your conjecture in part (b)?

6. Graph $y=2t^2$ and $y=2t^2+3$ together. Explain geometrically why the two functions should have the same derivative at every \(t\).

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