Collegeboard and the AP Calculus exam is done

[Ebolamonk3y]

Do not talk about the Mulitiple choice exam.

DO NOT TALK OR CONVEY OR ALLUDE TO THE MULTIPLE CHOICE PORTION OF THE EXAM AT ALL! Discussions of the Free response is permitted as of Friday or 48hours from 8AM Wednesday 5/5/04

That is not allowed because they recirculate that year after year. But how did everyone do on the Free response portion of both AB and BC exams? :)


I thought the BC no calc was hard as SHlT and the with calc was piece of cake.

 

[cookiemonster]

I've heard differing opinions about the calc test. Some people have said it was pretty easy and some others have said it was tough.

cookiemonster

 

[Ebolamonk3y]

:) good stuff.

 

[Ebolamonk3y]

Alright... Its officially 48hours after the AP Calculus BC Free response stuff...

How did everyone do?


I thought the Part II no Calculator part was harder than the 1st with calculator part...

Question one took the longest time with that differential equations. And I got 12 and 12 for the limits as t approaches infinity.

 

[Theelectricchild]

I took BC last year--- the added series part of the exam are only there to boost my grade.

I love series.

 

[Ebolamonk3y]

:) You a Ramanujan Jr?

 

[Inspector Gadget]

Has anyone worked out the answers to the AP Calc. AB free response problems yet? I'm trying to gauge how I did, but I don't know if what I'm doing is correct or not (obviously, or I wouldn't be asking for others' answers). They can be found at http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html for anyone wanting to look at them.

 

[cookiemonster]

Why don't you post what you did and we'll check 'em for ya', instead of us doing all the work you've done already all over again.

cookiemonster

 

[Ebolamonk3y]

http://apcentral.collegeboard.com/repository/ap04_frq_calculus_bc__36109.pdf

Try the BC... I did all these... 12 and 12 for one part in 5... don't know about the rest... dun get this remainder... :(

 

[Theelectricchild]

Looking at the second problem on this test, it angers me how my Calculus III course last quarter never went over Taylors Inequality and overall applications of Taylor Series...

 

[Ebolamonk3y]

Not a Calc III topic :) Its not that bad... Just take a look at a it for a while...

 

[Inspector Gadget]

For test AB, Form A, my answers...

1.
a.) 2474 cars (integral of F(t) from 0 to 30)
b.) Traffic flow is decreasing because F'(7) -1.873 < 0
c.) 81.899 cars/minute (1/(15-10) * integral of F(t) from 10 to 15)
d.) 1.518 cars/minute^2 ((F(15)-F(10))/(15-10))

2.
a.) 1.133 (integral of f(x) - g(x) from 0 to 1)
b.) 5.150 (integral of (2-g(x))^2 - (2-f(x))^2 from 0 to 1)
c.) no idea :-/

3.
a.) -.133
b.) Speed is increasing because a(2) < 0 [a(2) = -.133] and v(2) < 0 [v(2) = -.436].
c.) The particle reaches its highest point when velocity = 0 because the velocity function is the deriviative of the position function, resulting in a critical point where v(t) = 0. v(t) = 0 at time t=.433.
d.) NOT SURE BUT...The position at time t = 2 is -1.361, and is moving away from the origin because both its position and velocity is negative. (Position found by the integral of v(t) from 0 to 2, -1 because the position at time t = 0 is -1).

4.
a.) Just a proof, no problem there.
b.) The y coordinate is y = 2.
c.) The d2y/dx2 at point P is -2/7. Point P is a local minimum because -2/7 < 0.

5.
a.) g(0) = 9/2, g'(0) = 1
b.) Sort of a guess because of a lack of time: x = 2
c.) Didn't get to it.
d.) Didn't get to it.

6.
a.) [Slope field sketched].
b.) The slope is positive where y > 1 where x != 0.
c.) y = e^(x^3/3) + 2 [I think I was supposed to put the C as a constant infront of the e but was once again pressed for time and did it the fastest way possible.]

 

[Ebolamonk3y]

I noticed also that even AP teachers had much difficulty with this problem... On the AP Listserv they were always comparing answers to what is and what was because the answer keys are not posted just yet... There is, just like here, some complaint about somethings that regular Calc students should not get as a student, etc...

 

[kvzrock]

some error corrections:

2.
a.) 1.133 (integral of f(x) - g(x) from 0 to 1)
b.) 5.150 (integral of (2-g(x))^2 - (2-f(x))^2 from 0 to 1)
c.) no idea :-/

b) should be 5.15pi. Forumula for solids of rev. disk method : pi*integrate( (r1^2 - r2^2) dr )

4.
a.) Just a proof, no problem there.
b.) The y coordinate is y = 2.
c.) The d2y/dx2 at point P is -2/7. Point P is a local minimum because -2/7 < 0.

d2y/dx2=-2/7 <0 hence the curve is concave down therefore it is a local maximum

 

[lilartsychic]

hey, does anyone have the answers to the part II's for calc AB for form B

 

[mathwonk]

I am a university calculus teacher and the AP tests you referrred us to look very tedious to me. Here is the kind of test I have given my hoinoirs classes at nuiversity in calculus:

do they look easy to you? I am just curious.

2310H Test 2 Fall 2004, Smith NAME:
no calculators, good luck! (use the backs)
1. (a) Give the definition of "Lipschitz continuity" for a function f on an interval I.

(b) State a criterion for recognizing Lipschitz continuity in the case of a differentiable function f on an interval I.

(c) Determine which of the following functions is or is not Lipschitz continuous, and explain briefly why in each case.
(i) The function is f(x) = x^1/3, on the interval (0, ).

(ii) The function is G(x) = indefinite integral of [t], on the interval [0,10], (where [t] = "the greatest integer not greater than t", i.e. [t] = 0 for t in [0,1), [t] = 1 for t in [1,2), [t] = 2 for t in [2,3), etc...[t] = 9 for t in [9,10), [10] = 10.)

(iii) The function is h(x) = x + cos(x) on the interval (- inf,+inf ).

2. (i) State the "fundamental theorem of calculus", i.e. state the key properties of the indefinite integral function G(x) = indefinite integral of f from a to x, associated to an integrable function f on a closed bounded interval [a,b]. You may assume f is continuous everywhere on [a,b] if you wish.
(ii) Explain carefully why the definite integral of f from a to b, of a continuous function f, equals H(b)-H(a), whenever H is any "antiderivative" of f, i.e. whenever H'(x) = f(x) for all x in [a,b]. Justify the use of any theorems to which you appeal by verifying their hypotheses.
(iii) Is there a differentiable function G(x) with G'(x) = cos(1/[1+x^4])?
If so, give one, if not say why not.

3. Let S be the solid obtained by revolving the graph of y = e^x around the x-axis between x=0 and x=3. Define the moving volume function V(x) = that part of the volume of S lying between 0 and x. (draw a picture.)
(i) What is dV/dx = ?
(ii) Write an integral for the volume of S, and compute that volume.

4. The accompanying picture is supposed to be a pyramid of height H, with base a square of side B. Define a moving volume function V(x) = that part of the volume of the pyramid lying between the top of the pyramid, and a plane which is parallel to the base and at a distance x from the top.
(i) Find the derivative dV/dx. [Hint: By similarity, b/B = x/H.]
(ii) Find the volume V(H).
(iii) Make a conjecture about the volume of a pyramid of height H with base of any planar shape whatsoever, and base area B.

Either: Prove the FTC. from part 2(i), you may draw pictures and assume your f is monotone and continuous if you like.

 

[mathwonk]

[Recall that a lower bound for a sequence {sn}, n ≥ 1, is a number K such that for every n≥1, we have sn ≥ K; and that a greatest lower bound for {sn}, is a lower bound such that no larger number is a lower bound.]
Assuming that {sn} is a weakly monotone decreasing sequence of reals, and that L is a greatest lower bound for {sn}, prove {sn} converges to L.

2.(i) Use an appropriate comparison test for series with positive terms to verify the series for ex: 1 + x + x^2/2! + x^3/3! + ... , converges absolutely for any fixed x.


If a sequence {fn} of continuous functions on [a,b] converge in the sup norm to a continuous function f, then prove the indefinite integrals of the fn also converge in the sup norm to the indefinite integral of f.


a) If f is a continuous function on the reals, with f(1) = c > 0, what else must be checked to conclude that f(x) = c^x for all x?

b) If a,b are positive numbers, use the method above to prove that the function f(x) = a^xb^x, equals (ab)^x.


If f is defined by f(x) = 1/2 for 0 ≤ x < 1/2; f(x) = 1/4 for
1/2 ≤ x < 3/4; f(x) = 1/8 for 3/4 ≤ x < 7/8; ...; f(x) = 1/2^n for
(2^(n-1) - 1)/2^(n-1) ≤ x < (2^n -1)/2^n; and f(1) = 0, explain why f is integrable on [0,1], and compute the integral. (The FTC is of no use.)

X. We know any function f: R+-->R which is
(i) continuous, (ii) not always zero, and (iii) satisfies f(ax) = f(a) + f(x) for all a, x >0 is a “log” function. Using this, prove that f(x) = integral of 1/t from t=1 to t=x, is a log function, using appropriate theorems. [Hint: You will need to show f’ exists and then compare the derivatives of f(x) and f(ax).]

XI. We know the only function f such that (i) f is differentiable, (ii) f(0) = 1, and (ii) f’ = f, is e^x. Assuming an everywhere convergent power series is differentiable term by term, prove the sum of the powers x^n/n!, for n non negative, converges to e^x. [Hint: First prove it converges everywhere.]




these tests were given to incoming freshmen taking second semester calculus with AP backgrounds.

 

[mathwonk]

extra credit: prove e is irrational by showing

n(1 + 1 + 1/2! + 1/3!+...) cannot be an integer for any positive integer n.

 

[mathwonk]

thanks for showing us these tests. as a college teacher i am benefited by knowing what kind of questions AP students are being asked.

i myself do not consider these tests easy for the most part ok, in a few cases, like what value of y makes y' = (y-1)^2 g(x) equal to zero). But there is nothing deep in them. and they do not greatly resemble the kind of tests i give in college. this claim by the AP people is false.

my tests are both easier and harder. they involve easier calculations but harder ideas.

these AP tests have a very high schoolish feel to them, in that the difficulty is all in the details, not in the concepts. I.e. these are questions about elementary ideas which are made technically hard, by using complicated figures or functions, and involving many steps.

Asking you to integrate a function that has 5 different formulas for its definition over an interval is not my idea of an interesting problem. note however that my own test has a similar problem, but in mine, there is pattern to the different formulas, and the integral turns out to be the sum of a series.

moreover in mine the concept is deeper because my function is not continuous, and it helps to know that any monotone function is integrable.

these problems are typical high school contest problems in that they test cleverness and complicated calculations as opposed to depth of understanding. I remember these from my high school, math contests days.

this may be a good thing, as young kids find this kind of competition fun, but it is not true that it resembles what I want my college students to know. I was good at these trick questions, and I preferred it, as that was easier than learning the ideas by studying hard and thinking. I even won prizes at it, in fact I was state champ at some such stuff, but I was not ready for college math.

I want my students to understand what integration means, and which functions are integrable, and why, not just be able to calculate tricky integrals.

I might ask a beautiful and surprising integral, like proving the surface area cut from a sphere by two parallel planes depends only on the distance apart of the two planes.

it is well known that archimedes was proud of calculating the ratio of the volumes of a sphere and a cylinder containing it exactly. but it is also of interest that the surface area of the sphere is the same as the lateral surface area of the cylinder, and that in fact any two planes parallel to the ends of the cylinder cut the same area on cylinder and sphere.

it is also helpful to understand why archimedes was able to calculate the area under a parabola as well as the volume of a sphere, and also the sum of the squares of the first n integers,..they are all the same problem! do you see why?

they offer us college profs $1500 a week to come and grade these tests but after seeing one, I would not do that torture for much more money.

you should be proud of yourselves for being smart enough to do well on these tests, but do not be fooled into thinking good college courses are like these. Of course the more stduents come in with this background, the more we may alter our courses to accommodate them, but there are a lot of solid, deep, courses out there that involve proof, discovery, and hartder ideas. I hope you encounter them as you are likely among the future mathematicians and scientists.

 

[Cincinnatus]

I've seen you mention lipschitz continuity several times on this board mathwonk... what is it?

I was never taught anything about it.

 

[mathwonk]

lipschitz continuity means the ratios deltay/deltax are bounded on the whole interval, or equivalently for functions that have derivatives, the derivative is bounded.

that is the most important kind of continuity for integration. most books restrict their attention to continuious functuions for integration, but of course many discontinuous functions are integrable.

after all integration is about finding area, and anyone can find the area of a rectangle, and yet every year I have students tell me that the function with value 1 on the interval [0,2) and value 2 on the interval [2,3] is not integrable because it is not continuous. note that the graph is composed of rectangles so any moron can find the area.

That is the kind of nonsense that comes out of some AP calculus (and some university) instruction. I am as interested in having my students think as in having them become little calculators. after all a $10 calculator can outstrip anyone at the latter, so how wil they live if they are thoughtless automatons?now in order to understand riemann integration, not only does one want to understand which functions are integrable, but also what kind of functions can be integrals. we all know that the indefinitye integral of a continuous function is differentiable, but how many have noticed that the integral of any integrable function is always continuous, even if the original function was not?but more is true, an integral is always lipschitz continuous which is much stronger. if |f| is a bounded function, say by K on the interval [a,b], and G is its indefinite integral, then deltaG/deltax is also bounded by K. so G is actually lipschitz continuous, which is much stronger even than uniform continuity.
suppose you want to mimic the FTC for integrable, not just continuoius functions? i.e. if f is only integrable, it is obvious that the integral of f on [a,b] equals G(b)-G(a), where G is the indefinite integral if f, i.e. where G(x) is the integral of f from a to x. But how do we recognize G in order to make use of this?

I.e. in the usual FTC, where f is continuous, we look for a function G that is differentiablke and has derivative equal to f everywhere. But what if f is only integrable, e.g. monotone? How do we find G?

It turns out that integrable f's are continuous almost everywhere, so the G we seek will also be differentiable almost everywhere, with derivative equakl to f where f is continuous. But what else should we require of G?

Well of course it makes sense to ask for G to be continuous, but this si not enough!

 

[mathwonk]

I.e., there exist continuous functions with derivative equalk to f everywhere f is continuous, but which diffewr from the integral of f. The key is to require lipschitz continuity.

I.e. to recognize the indefinite integral G of an integrable f, lok for a lipschitz continuous G such that G is differentiable wherever f is continuous, and G'(c) = f(c) at such points.The point is this is what is needed to rpove the mean value theorem. I.e. what is needed to prove the FTC is the knowledge that a function with derivative zero everywhere is constant. But this si not true for functions whicha are differentiable only most ofthe time. But for a lipschitiz continuous function, having derivative equal to zero most of ttime is enough to force it to be constant. hence this is the right idea.
why would someone not mention lipschitz continuity in an integration course, when that is the concept that distinguishes integrals?

 

[mathwonk]

the upshot of not teaching this idea, is that 5 or 6 years later when students take real variables, and learn that a funtion is the indefinite inetgral of a lebesgue integrable function if and only if it is absolutely continuous, (a generalization of lipschitz continuity), they have no idea where this came from.

also when they study differentiable equations, and lipschitz vector fields emerge as the natural ones for which the solutions are unique, they have no clue why. i.e. the MVT is the key to uniqueness of soluition of equations like G'=f, G(0) = a.

thanks for the question. am i making any sense?

by the way i myself learned these ideas by teaching out of good books in honors calc, like Courant and John. it made me so happy to realize that after 40 years I had finally understood the claptrap about absolute continuity and the radon nikodym theorem that I was taught in graduate reals, i tried to re introduce these basic ideas into calculus at the beginning.back when riemann defined integration, he proved on the next page that a function is riemann integrable if and only if the set of discontinuities has measure zero, but the gap between this fact and the definition has now reached 6 or 8 years in school, and is almost reserved for math PhD candidates. (But you do get it in spivak's calc book.)

i still remember the shocked and happy look on one of my honors calc, post AP, students faces when he asked me, after we had noted that all monotone functions are integrable, exactly which functions are riemann integrable, and i told him.

 

[mathwonk]

now you might think a student can wait for these subtleties, but how long should they wait to realize that after taking a calculus course they should not claim that a rectangle has no area. it is this sort of thing that got me started.

the point is whether a course is about understanding ideas, and why?, or about making highly tedious calculations that any calculator can do better.

 

[mathwonk]

here is another reason to teach lipschitz continuity: it is easier and more basic than regular continuity, and implies it.

e.g. the fact that deltay < K delta x, makes it trivial to see that deltay gets small as deltax does, and easy for anyone to compute the delta that goes with a given epsilon, when doing the feared epsilonics that students find so hard.

I.e. if delta y changes no more than K times as fast as delta x, then obviously the right delta to choose is epsilon/K.

anyone can do this.

suppose we are trying to find the right delta to go with a given epsilon to prove continuity for the function f(x) = x^3, at say x = 2. just note that [x^3-a^3]/(x-a) = x^2 + ax + a^2, so for a and x on the interval [1,3] say, this is bounded by 3(3)^2 = 27. so just take delta < epsilon/27.i.e. almost all functions actually met in courses are in fact lipschitz continuous, so the method is available.moreover the ease of the comparison between deltax and delta y for these functions helps make the point about what continuity means, not that "you can draw the graph without pickling up your pencil."

that intermediate value version of continuity does not convey the essential apporoximation property of continuity so useful to physicists and applied mathematicians, and everyday persons, i.e. the fact that changing the inputs a little only changes the outputs a little.comments?

 

[mathwonk]

the problem with AP tests that I have noted here is exactly what worries me. My kids high school used to have lots of innovative and creative honors courses, where kids learned to think deeply and write persuasively.

But when some well intentioned but ignorant parents (including myself) argued and complained they did not have enough AP courses they changed. They phased out the good courses and replaced them with these mickey mouse, even if difficult for the wrong reasons, AP type courses, and now the parents are happy.

the kids are impressing admissions counselors, and getting into colleges, but I worry they cannot think or read and write any more as well as they could before, and I presume they are struggling in those courses and schools where the profs still require intelligent thought and articulate argumentation.

I saw the results in my own kids, as they dominated in college in subjects for which they were not AP prepared in high school, and struggled in those where they were.

No matter how tricky and difficult they are, AP courses and tests represent a dumbing down of learning. Maybe not in weak schools where the level was low, but definitely in good schools where the level was higher.

And even in weak schools they are going in the wrong direction, the emphasis is wrong. If a kid is good at multiplication, you do not help her by asking her to multiply longer and longer numbers, you ask her why multiplication is commutative?

 

[mathwonk]

another way to use the concept of lipschitz cont is to see why a differentiable function is continuous.

i.e. if deltay/deltax actualy has a limit at c, say equal to f'(c), then |deltay/deltax| is certainly bounded near c, say by say |f'(c)|+1.

so if we get close enough to c that |deltay/deltax| is less than |f'(c)|+1, then for |x-c| < epsilon/|f'(c)|+1, f(x) will be within epsilon of c.here is a comment on the problem 4 on the 2006 free resposne AP/AB test, that goofy graph with 5 different shapes.

Suppose we define G(x) as the integral of that function from 0 to x. then ask someone to prove that G is continuous say at x=20, by finding the epsilon that goes with a given arbitrary delta.

how easy is this? it is not as easy if they stop to compute the formula for the function.

but if they understand that calculus is about the relation between deltax and deltay, they may just ask thenmselves how close do they have to be to the point x=20, so that the area under that graph changes by less than epsilon. then since the graph is never higher than 15, it is clearly epsilon/15.

what if instead of asking about epsilons and deltas we just asked the student how much the area changes when x changes by delta, for that graph?when giving a test, ask yourself what you are testing? to get this question does a person need to understand the concept? the terminology? how to calculate? how to follow directions?

once when i was young and competitive, I assigned a problem from a famous book stating that a certain irreducible curve formula had only one singularity, at the origin. I instructed the students to compute the contribution to the arithmetic genus coming from that singularity and to deduce the genus of the curve. then i asked, what did they conclude?

some smart students used a different, quicker method from the one i specified, and concluded the genus was such and so. but i had laid a trap for them, designed to catch what I considered lazy students.

If they had actually done the calculation the way i said, they would have gotten a negative answer, and the right conclusion was that the book was wrong, and the curve was not irreducible after all. So it was partly a lesson in not believing everything you read, and doing your own work.

But it was also one upsmanship by me, so perhaps it was inappropriate, as what i was teaching was that if they did what i said, they would learn something. the students who felt they could outwit me, did not learn it.

I no longer do things like this as I do not want them losing their confidence in their cleverness, or deciding that I am always right. still it is difficult to teach students who do not think they will learn from doing what they are asked to do.

i recommend choosing good teachers, having faith in them, but never getting to the point where the only reason for believing something is the teacher said so.

 

[mathwonk]

ok i confess, i am posting so much because i am putting off grading exams, because grading is so much more odious than teaching. but you could probably figure that out.

 

[mathwonk]

heres another question about #4 on the AP/AB test. is the indefinite integral G differentiable at (20,15)? if so what is the derivative of G there?

does it have a second derivative there? if so what is it?

i claim these are much better, and easier, questions about that graph,they require less computation and test more understanding.

 

[Cincinnatus]

Is there a way to prove that a function is integrable iff its only discontinuous on a set of measure 0 without using all this lipschitz stuff?

Because I definitely remember learning that criterion in my multivariable calculus class. I don't recall how we arrived at it though. We definitely didn't use lipschitz continuity (at least not by name) since I'd never heard of it before...

 

[mathwonk]

sorry. i mixed two concepts, namely
1) how to characterize a function which is integrable, (one with a measure zeros et of discontinuitties),
and 2) how to characterize a function which is an integral, namely it is lipschitz continuous.

they are unrelated.

 

[mathwonk]

i seem to have made a simple concept look hard. lipschitz continuity is much easier than regular continuity.

i.e. it just says that for some K>0, the change in y is never more than K times the change in x.

this makes it very easy to check regular continuity, namely that small changes in x produce small changes in y.


do you know the epsilon delta definition of continutity? i.e. that for every epsilon, there must exist a delta such that when x changes by less than delta, then y changes by less than epsilon?


well how do you find that delta? say for f = cosine, at pi/6?


the lipschitz princiople tells you to look for a bound on the slope of f, namely 1, in this case, since the derivative is bounded by 1.

thus delta can always be taken as 1.epsilon!


how easy is that? did you do exercises where you were given an epsilon and had to find a delta?

if not you may not appreciate what this does for you, but if you did, you should.

 

[mathwonk]

on the other hand if you had a sophisticated course and learned about sets of measure zero, try this: prove: if f is lipschitz continuous on [a,b] and has derivative equal to zero except on a set of measure zero, then f is constant.