2024 Calc AB & Calc BC FRQ #1

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Free Response Question #1 from the 2024 AP Calculus AB and AP Calculus BC exam.
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Average rate of change; units of derivative; average value; Left Riemann sum; fundamental theorem of calculus (FTC); increasing or decreasing rate using second derivative

Пікірлер: 61

@Vrexell 2 ай бұрын

Ive found multiple other teachers solving this problem (1d) on social media and they say decreasing because you interpret the rates as an absolute value

@basketballpro2008 2 ай бұрын

For part d, the temperature should be changing at a decreasing rate because c’ and c” are in opposite directions I would think just like speed decreases when velocity and acceleration are in different directions.

@Vrexell 2 ай бұрын

This is exactly right and even physics agrees. A cup of coffee that is cooling will cool at a decreasing rate as it gets closer to room temperature

@luccohen4767 2 ай бұрын

For part D., would the rate not be decreasing, considering the fact that C'(t) is negative for this interval? Thus, the rate of change in temperature would be approaching 0, rather than speeding up.

@Han-D4ror2 2 ай бұрын

no, it would not be. there is nothing signifying that it is asking about magnitude only. it simply asked if the rate was increasing or decreasing, which it was increasing

@lovepng849 2 ай бұрын

omg been waiting for this thank you!

@shotsquad8255 2 ай бұрын

I feel like your explanation would only apply if the first derivative was also positive. Then the rate at which it is changing would be an increasing rate.

@zzgaming29 2 ай бұрын

I had a feeling it was some sort of trick question as I was writing increasing because it didn’t make sense that a rate that was becoming less negative was “increasing” since rates make more sense when interpreted in absolute value. Should’ve gone with my gut instead of keeping increasing.

@theschoolgy1985 2 ай бұрын

the rate itself is still increasing regardless of whether the first derivative is positive or not. if it’s positive it’s increasingly positive, if it’s negative it’s increasingly negative

@Vrexell 2 ай бұрын

@@theschoolgy1985no if f’ is negative and f’’ is positive then f’x is increasing but the question specifically asked about the rate at which the temperature changes so we need to use absolute value of f’. We can see that it gets closer to 0 so it is cooling at a slower rate (decreasing)

@shreddedtwopack6625 2 ай бұрын

@@Vrexellnah rate of change can be negative. You’re mixing velocity and speed.

@dank. 2 ай бұрын

@shreddedtwopack6625 I definitely think the college board is the one in the wrong here but I don't think "the rate at which the temperature changes" is the same as "the rate of change of the temperature". The temperature changing is a scalar quantity like speed, so it's positive. From there we just apply the argument that the signs of f' and f'' are different

@goat7309 2 ай бұрын

bro you were such a help in my studying i wish i had found your channel earlier in the class thank you for the great resources 🔥🔥

@turksvids 2 ай бұрын

Thanks so much! It would help out the channel a great deal if you shared with some future calc kids!

@icii7379 2 ай бұрын

for question 1 wouldn’t it have to be at a decreasing rate because c’(t) is always negative so it’s absolute value is decreasing over the interval because c’’(t) is positive?

@pablonerudaishidingincuba3657 2 ай бұрын

For 1d, I said the rate was decreasing, I might have misinterpreted the problem. In the question, the rate of change would be the absolute value of the derivative (since it is not making a distinction between increasing/decreasing temperature, only increasing/decreasing rate). If the absolute value of the derivative was decreasing, then the rate of change would be decreasing. If C’(t) is negative while C”(t) is positive, then |C’(t)| would be decreasing.

@turksvids 2 ай бұрын

I think it will be a big topic of discussion at the grading about this. I interpreted it as asking if the rate is increasing or decreasing. The rate, C', is increasing because C'' is positive. I can see your perspective as well. I look forward to seeing how the grading goes!

@pablonerudaishidingincuba3657 2 ай бұрын

yeah if you interpret “increasing” as closer to positive infinity (which I guess is the technical definition) your answer is definitely correct. However I feel like in context, interpreting “increasing” as a more significant change is also reasonable and very natural. Do you think it’s a possibility they would accept both interpretations? I’m not tryna lose points 😭😭 I really feel like they should have been more clear (e.g. distinction between speed vs. velocity in other problems), but idk

@fergi8089 2 ай бұрын

Since C'(t) is negative and C''(t) is positive, isn't the temperature changing at a decreasing rate for part (d)?

@imnobody101 2 ай бұрын

I feel like the first half of this year's FRQs were super similar to last year's FRQs which makes me feel good bc I spent most of my time studying the 2023 FRQs

@turksvids 2 ай бұрын

Nice! It always feels great when you happen to study problems that turn out similar. On the flip side, I spent a lot of time with my class going over FRQs that featured stuff that didn't even really show up here. MVT and Candidates Test most notably this year.

@Dollaboi792 2 ай бұрын

For 1b. If I had 985 written and the work for the Reimen sum but ended up dividing by 12 at the end to use in the interpretation would I still get the point for the approximation and the Reimen sum

@potatotree7143 2 ай бұрын

was literally cooked, so depressed now omg 😭

@nak_attak 2 ай бұрын

Def got full credit on this one

@speedster2360 2 ай бұрын

For part d, since the first derivative is negative and second derivative is positive, doesn't that mean the temp is going down at a decreasing rate?

@cheaponation3131 2 ай бұрын

Turk my goat having the same answer for d fills me with confidence.

@jonahmurovitz8454 2 ай бұрын

Since the first derivative was negative for all of the given values, and the second derivative was positive, shouldn’t the temperature be changing at a decreasing rate?

@muff1n.894 2 ай бұрын

cant wait for frq #6 🥲

@hannahoued6143 2 ай бұрын

oh i FLOPPED damnit

@moomoo-jt8yq 2 ай бұрын

For part d, wouldn’t the temperature of the coffee be changing at a decreasing rate because C’(t) is less than 0 and C’’(t) is greater than 0?

@andrewmorgan2241 2 ай бұрын

it is still changing at an increasing rate, but in the negative direction. It can increase negatively. Increasing is getting further away from 0

@accxlxrxtxr9107 2 ай бұрын

For part b did you have to solve for the average value? All I did was recognize that it was the average value and explained how it fit into the problem.

@accxlxrxtxr9107 2 ай бұрын

Also For part d I accidently said decreasing rate but I did put c’’(t) > 0, will I still get partial credit for that?

@glacius6860 2 ай бұрын

I got changing at a decreasing rate because the 2nd derivative is positive but the 1st derivative is negative

@Dollaboi792 2 ай бұрын

Second derivative being positive means first derivative is increasing

@dank. 2 ай бұрын

True! Much like other internet math arguments though, the dispute is over the imprecision of the problem in their defenition of "the rate at which the temperature changes".

@mogus6030 2 ай бұрын

These FRQs are different to the ones I got? Are there different sets?

@turksvids 2 ай бұрын

There are a couple different sets of FRQs that they use each year to account for time zones. (And I think to test for variations in difficulty, maybe, or to see if people are cheating within a school. I've definitely had students with different FRQs in the same class in the same year.)

@shotsquad8255 2 ай бұрын

I’m so lost why d on BC isn’t decreasing. It’s asking about the rate at which the coffee is changing. So If the first derivative is negative from 12 to 20 that means that the coffee temp is decreasing but if the 2nd derivative is positive then that means that the first derivative is has a positive slope and bc it’s negative that means it’s getting less negative from 12 to 20. If the first derivative gets less negative doesn’t that mean the temp of the coffee is changing at a slower rate now? The 1st derivative at t=12 is about -2.3 and at t=20 it’s about -1.5. That means it’s now cooling slower than it was prevously so the rate at which is it changing is decreasing

@JashXD 2 ай бұрын

ye alot of people are saying that and i agree

@Han-D4ror2 2 ай бұрын

the rate is increasing in that it is becoming less negative.

@firingsquadgeneral 2 ай бұрын

For part D, dont you have to compare the signs of C'(t) with C''(t)? Since C'(t) was negative and C''(t) was positive, the first and second derivatives have opposite signs and therefore the rate would be decreasing, right?

@rndm4265 2 ай бұрын

I forgot to add C(12) for part c😭

@stephen7469 2 ай бұрын

I disagree with your 1d.) the problem is asking about the temperature of the coffee and the rate of the coffee. For 12

@stephen7469 2 ай бұрын

Your response is saying that the rate of change of the coffee is positive (true), but the question is asking us about the temperature of the coffee, NOT THE RATE OF CHANGE OF THE RATE OF CHANGE, and I could be completely wrong, but thats why i think its decreasing because the question was about the temperature, not C'(t).

@winter5017 2 ай бұрын

@@stephen7469 This has to do with speed and particle motion. There are no particles here, thus the video is correct. Although I admit it is worded really weirdly and makes you think it's speed but it's not.

@pablonerudaishidingincuba3657 2 ай бұрын

I agree with you, I said the same thing for my frq. In the question, the rate of change would be the absolute value of the derivative (since it is not making a distinction between increasing/decreasing temperature, only increasing/decreasing rate). If the absolute value of the derivative was decreasing, then the rate of change would be decreasing. If C’(t) is negative while C”(t) is positive, then |C’(t)| would be decreasing.

@Vrexell 2 ай бұрын

@@winter5017physics disagrees with the videos answer tho. Why would a cup of coffee cool faster the longer it sits out. As a mass cools, the rate at which it cools slows down as it approaches room temperature

@turksvids 2 ай бұрын

i agree with you agreeing with me, but i definitely see some compelling arguments for interpreting it the other way. i'd still say the rate is increasing.

@yobamayobama8940 2 ай бұрын

Will they ever release form e

@turksvids 2 ай бұрын

unfotunately they only release one set of FRQs per year now, so those are locked in some kind of vault or something for the rest of time...

@CrimsonDragon0801 2 ай бұрын

Noo it was 3 intervals? WHY DID I DO 4? AGHHHH! I THINK I DID 4 IM NOT SURE NOW...

@JashXD 2 ай бұрын

how is 1d at an increasing rate if C’0

@JashXD 2 ай бұрын

if a graph is decreasing concave up u wouldn’t say that it’s changing at an increasing rate. i think this problem is analogous to speed inc/dec

@turksvids 2 ай бұрын

i actually would say it's changing at an increasing rate in that case. say that c' is -10 and then c' is -1, did c' increase or decrease? (or in this case i think you can literally maybe look at the graph of c' to see that c' is increasing.) the question is definitely worded in a kind of tricky way, though, so we'll see what the official scoring says!

@Vrexell 2 ай бұрын

@@turksvidsthe derivative did increase but it doesn’t ask about the derivative. it says “the rate at which the temperature is changing” which brings in the absolute value

@xeroc4520 2 ай бұрын

damnit i said decreasing im stupid

@Vrexell 2 ай бұрын

You’re right it is decreasing. Look at everyone else’s comments about part D. F’ and f’’ have opposite signs and its absolute value so the rate at which it is changing is decreasing

@RebornKaotic 2 ай бұрын

@@Vrexellno

@dank. 2 ай бұрын

I think your takeaway should be that college board is the villain lol