Revisiting 2024 AP Calculus AB FRQ #1 (Part D)

So this has been a HUGE discussion. I just want to give you my two cents on how I tend to view rates and interpretation, but it may not be the same as yours.

But this is probably easier to convey than the dozens of messages I've sent so far on it.

[patrickfinn]

If I said "increasing function or decreasing function, " no one would have an issue with agreeing that you should determine if the first derivative of the function is positive or negative. Replace the word "function" with "rate" and you still do the same thing - if the first derivative of the rate (aka C" in this problem) is positive, then the rate is increasing; if the first derivative of the rate is negative, then the rate is decreasing. C"(t) > 0 so C'(t) is increasing so the temperature of the coffee is changing at an increasing rate.

You can also look at it this way - Increasing rate means that every answer for the rate on the given interval is greater than the proceding answer so the graph will go "up" from left to right. When you graph C'(t) on the interval from 12<t<20, you will clearly see the graph of C'(t) goes up therefore C'(t) is increasing which means C'(t) is an increasing function which means the temperature of the coffee is changing at an increasing rate.

Ultimately, I agree the wording is a little tricky but you have to focus on what it means for a particular function to be increasing or decreasing.

Thank you for all the videos with your work as well as the explainations.

[snalchbalfus]

In my eyes, the coffee’s temperature is decreasing because c’(t) is negative and c’’(t) is positive on the interval. So the graph of c(t) will be going down but it is concave up, meaning the slope is down, but it is getting less negative. So I believe the answer is that the temperature is decreasing at a decreasing rate

[rospore]

When I see the words, "The temperature of the coffee is changing", to me that means that either the temperature is increasing OR decreasing, doesn't matter which. So I interpret this as the "speed" in your physics scenario, not the "velocity". Since the second derivative is positive and the first derivative is negative, that means the rate of change of the temperature is approaching 0. Therefore I don't really think it would make sense to say "The temperature of the coffee is changing at a increasing rate", when the rate is literally approaching zero and getting slower and slower. If the temperature really was "changing at an increasing rate", then as time goes on you should expect to see larger changes in the temperature over the same time interval, which isn't happening in this case as instead, those changes in temperature are getting smaller and smaller. TBH though, I think we can all agree that this problem is just terribly worded and I hope collegeboard will give points for presenting either case.

[SM-ghcy]

Solutions are out and I still don't agree with it.

[myusername]

When you graph C"(t), it is positive for 12<t<20, so wouldn't that just mean that C is going at an increasing rate? (positive 2nd derivative means concave up, which means the rate is increasing)