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

Activity 5

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

1. Use your graphing tool — a calculator or a computer tool — to determine the values of `ds//dt` at `t = 1`, $\mathrm{`t}=\mathrm{2`,}$ and $\mathrm{`t}=\mathrm{3`.}$

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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