Oxford University Mathematician takes American AP Calculus BC Math Exam
University of Oxford Mathematician Dr Tom Crawford sits the AP Calculus BC exam with no preparation. The exam is often taken by American High School students as part of college application process.
There are 5 AP Math exams, 4 on Calculus, and 1 on Statistics. The Calculus BC exam is the highest level offered.
The paper in the video is the 2022 version of the exam, Section II, Part B (non-calculator). You can download the exam paper and mark scheme for yourself at the links below.
BC Paper Section II: tomrocksmaths.files.wordpress...
BC Mark Scheme Section II: tomrocksmaths.files.wordpress...
The exam paper and all questions contained within are the property of AP Central. Material is used under a 'fair use' policy for educational purposes.
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: www.conted.ox.ac.uk/profiles/...
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Video edited by Ryd Cook.
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Please make video on JEE advanced solve (India)Math section 2016 exam .
Inflection is when gradient changes from decreasing to increasing. Remember 2nd deriv is the "gradient of the gradient" so when the "gradient" has either a min or a max you have inflection. Gradient does not need to ever be zero.
Hello Sir, u made an misact at #1:53 its indeed continuous. Cheers
He made a mistake a minute in. He labeled something as (4, 0) instead of (0, 4). Didn't lead to any miscalculations but imagine plugging those numbers in without realizing it. Actual mathematician doing the same stuff we do. That actually feels great.
he fixed it
0:33 Quick correction: Calculas AB and BC are the 2 calculus courses, Precalculus isn't really a calculus course, more just expanding on and connecting topics covered in algebra 2 and geometry and Adopt AP Precalculus isn't a course but it's there because this is the first year that AP Precalculus exists so it contains information for schools and stuff to learn about it.
Isn’t ap multivariable calculus a thing
@@VoGnos no multi is pretty much exclusively a college/dual enrollment class for hs
So what was the point of that
@@VoGnos Multivariate calculus was the fourth semester in my undergraduate school. This covers the first three semesters.
Just started watching your video, the one on GCSE exam paper. I’m glad to see I have more to watch! These are incredibly fun to watch
You gotta do the 2023 frq that’s the exact test I wrote!!!
Whaat you wrote it?
Same here. I didn’t find it difficult except for the last one, but I think that I (probably) got it right
@@8ball708 i think he means thats the one he took
@@stuv12 Ohh that makes more sense
Bro u evil
As a student studying for calc bc, this video was phenomenal and fun to watch
I had this exam when taking AP Calc BC a couple years ago! I still remember that 6b problem because the 1/10 tripped me up haha. Loved seeing you go through the exam!
Wow I’ve been waiting for SOMEONE to take the BC exam, and you’re the one. Ive been straight up practicing for a year already.
I didn't actually take the material in the third semester of Calculus it covers... I did it independent study. The former calculus teacher, who had been promoted to vice principal, really understood Calculus and he helped me with that work. My high school was little more than a prison for the students unfortunate enough to be assigned to it. Without that VP, I would have been expelled.
I love you man , God bless you and your country 🇬🇧❤
I remember taking AP chem in high school. That course had me on my knees. Watching you work through the FRQ practice problems gave me flashbacks.
I'm taking ap chem right now and I hate it lol
@@bb4251 the exam was brutal 💀I took calc BC today and it was moderate
I took my AP Calc BC test back in 2022, one of my favorite AP classes I ever took. Got a 5 in it :)
Nice work getting a 5.
A 5 is a good attempt especially on your first try, don’t beat yourself up too much. You can always retake and reach for that perfect 10!
@@marving8907 I believe 5 is the maximum score:)
@@mathsunmasked56 yeaa I’m just messing with yea lol
nicely done
I’m not near any of this but the fact a person this established is working this hard and thinking this hard ab this scares me
😂😂
bro ts is easy asf ur finished
It just proves that the way we teach calculus to high schoolers is a bit backwards. Students who are able to perform well on these exams typically just study these problem-specific methods of answering a question. This man is a doctor in his field and is relying purely on knowledge of mathematical concepts to complete the exam. I guarantee that the first problem has some "simple" method that students are taught to rote memorize, when in reality this type of learning is unhelpful for most mathematicians.
I love seeing this so much because as I do math I have to have all these same thoughts. And now that im a teacher it's great to see a professor struggle with these same things.
Agreed, in my experience the best professors generally weren't always the ones who had maths come super easy to them, it's through their struggles and perseverance with mathematics that they are able to break things down in a simple intuitive process. Richard Feynman is a perfect example of this.
You're unnaturally stylish for a mathematician, Dr Tom! ;-) Keep up the good work sir, much love, much love...
Maybe 20 years ago lmao
@@krissv3ctor512 he did stay stylish, rather than fashionable.
Not at all? He looks like a younger version of every math professor I ever had, dresses exactly the same
@@AnthonyLauder fair enough
As someone taking AP Calculus BC right now, this was very fun to watch
Same
Me dreading and excited for the exam as a junior in high school 😢
There’s no IB math HL at my school (only SL), so I’m doing this next year. Ty for making this
Love this video
As an AP Calculuc BC teacher, I really enjoyed hearing your thought processes.
Agreed
Love your videos, I was wondering if you could make a video about tetration,pentation and hexatation.
for the first question. you needed f(4) because of the integration constant. there would be no way of knowing any value of f by knowing f’ without knowing atleast one value of f at a certain point. the higher the order of derivative the more unknown constants, and therefore we’d need a value of f and f’ at a certain point to know anything about f and f’ from f’’
Awesome vid! As an ap calc tutor for a good several years, I can say that the exam right after the covid/lockdown years were certainly a bit nerfed. The ability to analyze graphs has suffered in the states and I think the test has slightly altered in acknowledgement of that
thanks, we love your lectures. Would you also do the same for IB DP Math AA (HL) calculus ?
Thanks for doing this! It was fun to watch. Great job on that part of the BC exam! Regarding the endpoints, I had totally forgotten about checking them as well (darn those endpoints!). For the marks, I would not have counted Part d of the last question because you would not had time to actually do Part d if you were doing those questions for real. So I'd treat it as you getting 33/36, still an excellent score. I took the AP BC exam about 35 years ago. Back then, each student who took the exam got a score from 1 to 5 where 1 was basically an F and 5 was like an A/A+. If what you did was the entire exam, I think you would have been given a 5. You needed at least a 3 in order for the high school calculus classes you took to be considered for college credit but I think you would have had to get a 4 or 5 (and not a 3) for most universities to consider giving you credit for that calculus work.
I don't think there's much of a point in being pedantic about timings because he spent so much time explaining what he was doing in reality if he was doing this in a test he wouldn't run out of time
they still use the 1-5 scale today
In a real exam he would have been told to leave the exam hall for constantly talking and giving away answers to other students. So 0/36 if we are being realistic.
Teaching AP cal this year, i would have scored a practice like yours as a 32/36 - you equivicated on the two problems where i would mark for notation. This is still well on pace to get a 5 on the test, which is the highest score.
Love doing these exams; keeps me fresh
Hi there. I really like your videos. I like maths. I do not know, if someone already told you but the german Abitur is spelled without the first r, it is Abitur.
Do the 2024 FRQ when it comes out in a day or two, that thing you should easily get a 4/5 in, but some of those non-calc frq questions were rough
AAAHH THIS IS THE TEST I TOOK!!! brought back memories fs haha (hoping i make it into oxford next year!!)
when finding the radius of convergence by ratio test, it does not tell you what happens at the end points, those have to be tested separately.
Hey Tom love your videos! I was wondering if you could perhaps do a video where you take the JEE Advanced Math exam from the entrance exam into the IITs. Thanks!
on b on the first question, an inflection point has to have a second derivative value of 0 and a change of sign, so your reasoning can just be that on the first derivative, the graph at x = 2 has a gradient of 0 and changes sign on either side, therefore the inflection point is at x = 2
Good luck with the ap exam 😊😊
Taking this this year. I hope I do well
Please do the IB HL Analysis and Approaches papers
At 1:17:00, im pretty sure you would only get 1/2 since the scoring notes say that you have to state that the series is alternating with terms decreasing in magnitude to 0
Hi Dr. Tom! As someone who did calculus in the US, I’ve never heard of your (sorry if I misspell it) quadrature method (30:28)that is better than Riemann sums. What exactly does the quadrature process entail (I imagine something to do with “four”)? Thanks; great video so far!
"Quadrature" is a historical term that just referred to a geometrical method used to approximate an integral/area. Some mathematicians would actually consider a Riemann sum to be a (very simple) type of "quadrature" for estimating an integral. Usually, however, numerical methods classes would start with something a little more sophisticated to estimate area as a "quadrature" method. He mentions the trapezium/trapezoidal rule, which is sometimes taught in American intro calculus classes, where you draw a trapezoid and find its area for each Riemann sum interval, rather than using a rectangle. (That is, the two endpoints of the interval on the function are used to construct a trapezoid for each interval -- you'd sum the area of those trapezoids.) Tom also mentioned Simpson's rule, which basically would use a quadratic function (parabola shape) to approximate the shape of the function in each interval (rather than a rectangle or trapezoid). You'd find the area under all of these parabolic segments and sum them to approximate overall integral. Quadrature methods have other more advanced approximations for numerical integration too that focus on optimizing the calculations using increasingly better approximations to the functions within the intervals. There are whole classes of approximation functions designed for this sort of thing. The word "quadrature" itself, as you guessed, does have something to do with "four," as one of the first geometrical problems these approximation methods were applied to was trying to "square the circle" in ancient Greek mathematics. That is, to construct a square (a quadrilateral, hence quadrature) that would have the exact same area as a given circle. Numerical integration methods begin with using other quadrilaterals like rectangles and trapezoids to approximate the area under a given curve, so that's the etymological root of the term. However, as I mentioned above, in modern usage "quadrature" often just refers to any numerical approximation method used to find an area under a curve (i.e., an integral), whether it uses quadrilaterals or more sophisticated shapes.
How do you find f() from f'() ??? By integration , ie just find the area under the f' function.
i was thinking of integrating sine shifted 180 degrees with theta multiplied by pi/4 over 0 to 4 then subtracting it from 3 for the circle part. since derivative is below axis you wold expect y to be higher than 3 at the orgin... i guess that wouldn't work though.
AP calc bc is calc 2 right? One of the hardest classes I’ve taken imo
This is the nasty exam which covers three semester of college Maths. Because of the time limits, it was the hardest exam I've ever taken. The essay section (Part II) is NOT expected to be finished. Only an exceptional Maths student would even come close to finishing it. My undergraduate school's Dean told me I had come as close as anyone he'd seen and made no mistakes on it at all.
What is the green triangle tattoo on the left arm down near the elbow? I've seen this symbol before. I hope it's not that.
the point of inflection can be calculated by differentiating the semi circular path of f’ to give f’’ for 0
so x=6 technically isn’t a solution because f’’ goes from 1 to -1 but never any value inbetween. simply because the function is piecewise
you should do the Aus HSC math extension 2 paper
Which application are you using to to solve those questions on your tablet?
I'm chuffed... chuffed to see a god tier oxford math prof does calculus like I do.
Do leaving cert (Irish end of secondary school exams) maths and applied maths too
Love the vid dude, awesome to hear your thought process in real time. Correct me if I’m wrong but I think at 1:09:58 you gave yourself full marks however the interval should’ve been closed at 5?
Calculus was my strong point but I've not really used it for 30 years so it's easy to forget some aspects. That Q3 was far from easy. Point of inflection is where f'' = 0 as I recall. Looking forward to next questions.
perfect illustration of my "super-power" when I write this exact same exam - being able to picture the problem in my mind - that is a gift from god and no amount of practice can replace it.
I think the 2023 test was a bit more challenging than other tests
42:33 why isn't he multiplying the 1/x with xe^(x/5) and THEN integrating that?
How long into the video until he corrects the equations of the lines?
Anyone know what application Tom uses for writing things down?
Do the GRE math part for the next video
What is the name of the app he uses on iPad to solve the exercises?
Isn't an inflection point changes from concave up to concave down and vice-versa?
Yes! It is where the acceleration flips signs +/-, giving it a changing concave up and down shape
You can use calculators on it now? Back when I took it it was all concepts and symbol manipulation and no hard calculation that required a calculator, much less a graphing one.
There is a calculator and no section. Calculator is not required but it makes calculations faster.
there's 2 sections. AP Chemistry allows calculator now too but they added a bunch of question and topics that require calculations
Shouldn't you have used Bessel's Formula for the interpolation on Q4 part A?
Yes. True indeed. Bessel, not to be mistaken with Brussel...
OXFORD professor, please do a video with solving the CHINESE college entrance test, it is called GAOKAO, i am pretty excited about the video about that!
不可能，因为高考好难呀😂
what do you think of math as a pre med major?
I took the AP Calc BC exam back in 2012, and I must say these questions seem somewhat familiar.... Edit: So I just looked up the questions from the 2012 exam, and they are in fact different. I did not realize they had them by year. Whoops!
What app is that
Please take jee main 2024 mathematics section ❤
You need to take off the marks from the bit where you were over the 1 hour time limit.
I still want to see the IB Maths AA instead of Maths AI exam cause maths AA is harder
You could have just found the midpoint between those 2 coordinates, or am I wrong?
3 a btw
42:50 I wholeheartedly agree; this question is WEIRD! I’m trying to figure out what the test-makers intend for us to do before you explain your understanding/reasoning on screen… this is the only one where I’ve been totally lost as to what their asking this far.
43:20 Your “MEH” here is amazing. I will suggest you’re able to have that attitude since you presumably know most (if not all) of this info already) and this is a test “for fun.” For those of us that had to take a test like this, this was traumatic in a way!
Tom, you've clearly never been to a banker's party in Miami. Plenty of ice sculptures there with a particular purpose!!😂
do BAC maths exams
Final question part d was done out of time?
I think he gets a bit of a break because he's talking to the camera and pausing to explain things to us while taking the test.
You should try Harvard's Math 55 papers.
You were too strict grading yourself. You only lost 1 point for missing the endpoints, but did get the final point for the interval. (It's an American marking scheme...)
Are there really that many ice sculptures in the US? Alaska has entered the chat.
You should try the asian college entrance exams like the gaokao and the IIT JEE maths sections. I can gurantee that they are harder than A-level FMs
Watching the first part of your video, it seems you are really rusty on your A-level analysis? You must do some really esoteric mathematics to have forgotten some of the basics like a point of inflection occurring when the second derivative of a function is zero?
Units!!!!!!!!!! in all calculations the ice sculpture task. Those mathematicians :)
What is the app hes using
It’s called ‘Paper’ on iOS
@@TomRocksMaths the man the myth the legend himself, ty man.
y=-x line is this ????????
algebra is all u need for a rpg like diablo
I passed my test . I farted , clear out the room , copy the answers from the smart kid in the class.
Do 1983. That's the one I took.
Lets play a game, shall we? How many times does Tom smack his lips?
How many A levels do kids take? BC is easier but the average US HS student takes 8 classes
3, 4 if high ability. 5 and up are allowed but very rare. There's one girl taking over 30 this year though!
Most top us students take like 15 of these ap classes lol, some take 29
@@blabbla-jp2snTaking more than 9-10 AP classes requires self studying and finding a school in the area that even offers the other AP exams. Plenty of top students take every AP they can fit into their schedule and no more, I took 8 and ended up with a merit scholarship to a top school without needing to take every ap exam.
I wish I was good at maths :(
Same
practice
Who else skipped to the end to see his score
Seeing him "struggle" makes this all the more interesting, and even moreso when you consider that some 15 year old nerd probably nailed this in half the allocated time 😂😂😂
Is AP like the old Advanced A Level (whatever that was called ?)
its the exact equivalent to the current A-Levels.
Regular High School courses in 11th and 12th grade in the US are less rigorous than IB, AP and A Levels. So the average high school student in the US actually comes out with a much poorer education then probably most of the economically developed world, and probably most of the world in general. This is important to notebecause in your question you assume that high school students in the US have the same rigorous curriculum as they do in England so advanced in your mind must be an advanced version of that but these people actually get College credit for first year courses for taking AP which is a really dumb system. Just make the high school curriculum more rigorous or have something at the level of AP or IB be the standard and then teach the people properly in University and don't spread out basic single variable and multivariable calculus over 3 semesters
This is an ironically uneducated post, filled with misconceptions, assumptions, arrogance and bitterness. Did an American girl break your heart? Were you rejected by an elite American university? @@Andalusian_
@@Andalusian_ have you taken any US high school courses
@@connorsheerin7563I am a bilingual IB diploma graduate with a bachelor's degree. I only took high school courses up to 10th grade, because after that I had IB. And we had Americans / a person who wanted to study in the U.S. in 10th grade classes because IB was too advanced for high school diploma or IB certificate people - they took the less rigorous course, which was the 10th grade course. Have you taken IB or A-Levels?
It’s so Sad that AP Calculus doesn’t teach Partial Derivatives I trade option derivatives using the black Scholes partial differential equation :)
a quant would never say something like that
I wish the problem you were working on would have been visible throughout the entirety of the video. It made my head hurt watching this.
This guy is a mathematician at Oxford? Seems pretty clueless.
54:35 this is multiple days in class yet Tom is just walking through it
Putnam exam next
Try IIT JEE advanced
Sir can I please get a heart big fan ❤
am I going crazy or is this a lightyears easier than A-level calculus questions... I mean the content looks the same but these question are really easy
they made questions easier for 2021/2022/2023 due to decreased scores from quarantine. they're making it harder again - but it still may be easier than a levels.
@@bb4251it's on par with or more advanced than A levels. The stats exam is more rigorous than the ALevel module counterpart.
Do HSC maths ext 2 exam
Emo mathematician 😢
is this how a professor of Oxford looks like?? freaking Oxford?? well yeah my desire to go to Across the pond just dropped to negative after watching this guy with tons of tattos and all
They don’t want you 😭🫶
Who else heard about this guy from @RHMathematics
I was wondering if barbers were on strike in the UK when this video was made. Also, it appears that combs are a rationed commodity. In any event, you might be successful in winning the Boris Johnson look-alike contest.