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Hi everyone I am confused on what the units of this derivative would be. The question is: . You are running on a treadmill, and your heart rate (R, in beats per minute) is a function of your speed (v, in km/h). Denote this relationship R = f(v). (a) Suppose f′(5) = 10. State the units of measurement for the numbers 5 and 10. (Clearly indicate which is which.) (b) Still assuming f'(5) = 10, explain the meaning of this statement in ordinary terms (that is, in terms of heart rate and speed)

For a I think the answer is beats per minute/ (km/h). Is this correct? For b I don't understand how to write the sentence becuase of the derivative. if it was just f(v) I would say, your heart beat is 10beats per minute when running 5km/h. Would I need to somehow convert the bpm to bph? Sorry for the long question

baroquefurniture
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  • If $x$ is in units $a$ and $f(x)$ is in units $b$, then $f'(x)$ has units $b/a$ because of the fraction in the definition of the derivative $f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{(x+h)-x}$. So it should be something like, weirdly enough, (beats per minute) per (kilometers per hour), or $\frac{\text{beats} \cdot h}{km\cdot \text{min}}$. edit: I didn't read your original guess but yes it's correct. – Keshav Oct 26 '23 at 22:30
  • @Keshav agreed. Units of derivative are # beats $~\div~$ distance. – user2661923 Oct 26 '23 at 22:42
  • If say your heart rate at 4.9 km/h was 109 bpm and at 5.0 km/h was 110 bpm and at 5.1 km/h was 111 bpm then you might say that running at a speed of around 5 km/h each extra 0.1 km/h added about 1 bpm, which you might describe as a rate of increase of 10 bpm for each additional km/h at this speed. Don't say this was an extra 600 bph per extra 1 km/h and cancel the hours to say 600 beats per km - this could easily be confused with the different calculation of 6600 beats in an hour in which you run 5 km so an average of 1320 beats per km at this speed. – Henry Oct 27 '23 at 00:17
  • @Henry Hi Henry, I am not sure what you mean. Because we are dealing with the derivative would we say the person's heart rate is increasing 10 bpm for every 5km/h or would it be something like the person heart is beating 10bpm as they are running at 5km/h? Sorry for the confusion, I appreciate your comment – baroquefurniture Oct 27 '23 at 03:18
  • @baroquefurniture Neither of those: closer to the person heart's beating $10$bpm more for each extra $1$km/h when they are running at around $5$km/h – Henry Oct 27 '23 at 08:09

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