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Handout Tuesday 9/9
Now we know how to approximate and compute derivatives (even if we don’t yet have shortcut rules). But what is the derivative good for? Why do we care? Sure, instantaneous velocity is great, but is it good for anything else?
There are lots of other “speeds” we can consider.
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Let \(f(t)\) be the temperature (in degrees Fahrenheit) in a room at time \(t\) (in hours). Then \(f'(t)\) is the instantaneous rate of change of temperature with respect to time, measured in degrees per hour. What does \(f(3) = 72\) mean? What does \(f'(3) = -4\) mean? Can that tell us what \(f(4)\) is?
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Let \(p(t)\) be the population of Greeley \(t\) years after 2020. What does \(p(5) = 120000\) mean? What does \(p'(5) = 3000\) mean, and what are its units? Can that tell us what \(p(6)\) is?
The units of the derivative are always \(\frac{units\ of\ f(x)}{units\ of\ x}\text{.}\) One notation for the derivative is \(f'(x)\text{.}\) Another notation is \(\frac{df}{dx}\text{,}\) which is read as “the derivative of \(f\) with respect to \(x\)”. This notation emphasizes the units of the derivative.
The derivative does not always have to be with respect to time. The preview activity for today asked about the function \(T(x) = \frac{50}{x}\text{,}\) which gave the time (in hours) it takes to drive 50 miles at \(x\) miles per hour. What are the units of \(T(x)\text{?}\) What are the units of \(T'(x)\text{?}\) What does \(T'(60) = -\frac{1}{72}\) mean?
