1. May 3, 2005

### ktpr2

If I get a B in Calc I but get an A in Calc II is my transcript doomed? I have a 4.0 but I'm a freshmen. Will it look a lot worse compared to someone with an A in both Calc I and Calc II or it wouldn't matter? (provided all my other grades are As).

2. May 3, 2005

### whozum

Don't even worry about it. Getting an A in Calc II is MUCH harder than an A in Calc I, so getting an A in the latter obviously proves you know your material.

3. May 3, 2005

### gravenewworld

For the last time people, your life is not ove if you get a B!!!!!!!!!

There are plenty of people who got B's or probably even C's who got into MIT, Harvard, Caltech, etc. which is probably what you are aiming to do right? Your transcript isn't everything.

4. May 3, 2005

### neurocomp2003

FIRST YEAR MEANS NOTHINGS>..its your last 2 years taht COUNT

5. May 3, 2005

### ktpr2

Thanks for all the insight. I suspected as such ... I'll budget my time far far in advance so I can get that A in calc II.

6. May 3, 2005

### mathwonk

let me ask you a few simple questions. So you got a B in calc I. Please state (without referring to book, other people, or notes)

1) the definition of a derivative.

2) the product rule, with hypotheses,

3) chain rule, with full hypotheses.

4) the mean value theorem.

now prove either 2, 3, or 4, and use the definition of a derivative to show directly that the derivative of sqrt(x+3) equals 1/[2sqrt(x+3)].

5) Now prove that a function f whose derivative equals itself, must be a constant multiple of e^x. hint: show f/e^x is constant, using the usual calculus method.

6) maximize the volume of a right ciurcular cone whose "slant height" is 10.

7) graph the function x/e^x, indicating all critical points, local maxs, mins, and flexes, and behavior at infinity.

8) compute the derivative of x^(ln(x)).

After answering these questions I will give you a grade in (non honors) calc I, comparable to a course in an average state university, if you wish.

7. May 4, 2005

### jma2001

What do you mean, "with full hypotheses"? Are you looking for a proof of the chain rule? Or something else?

8. May 4, 2005

### Muzza

No, the hypotheses are the conditions under which the theorem in question holds.

9. May 4, 2005

### mathwonk

i, mean, do not simply say the derivative of a composite is the product of the derivatives, say " if f is differentiable at a, and g is differentiable at f(a), then gof is differentiable at a and the derivative is...."

10. May 4, 2005

### jma2001

OK, thanks for clarifying that. My calc book doesn't go into that level of detail, it just says, here's the chain rule, now go solve some problems with it. Guess I need to find a better book.

Oops, I just checked one of my other books, and it gives definitions like the one you cited, so maybe I should use that one instead.

Double oops, you said not to refer to any books, so I'm sunk already.

11. May 9, 2005

### mathwonk

well you already get the idea of the level of your present course and knowledge.

12. May 9, 2005

### jma2001

mathwonk,

Believe me, I have no illusions about my present level of knowledge. One of the reasons I hang out here is to listen to experts such as yourself, giving me some idea of how much more there is to be learned.

I am curious, however, why you place so much emphasis on being able to state definitions and proofs from memory? Do you feel that a student of calculus does not truly understand the subject without knowing these things cold?

It is easy for someone who has lived with a technical subject their entire life to forget how non-intuitive it can be for beginners. To take myself as an example, I program computers for a living. Perhaps this is a bad analogy, but I think computer languages like C and Java have something in common with the language of mathematics. That is, in order to express an idea in computer language, you have to know a lot of details about the syntax of the language, the functions that it supports, and what kinds of statements are valid. When I was first learning C, I kept several thick reference books nearby at all times, and I referred to them constantly. Many years later, I have finally gotten to a point where I can write a modest program without referring to a book, but that is only a result of years of practical experience, not any conscious attempt to memorize the details of the language.

To take another example from my own experience: when I took statistics in college, all of our exams were open book, open notes. The trade-off was that the problems on the exams were much more challenging, and there was just barely enough time to finish them, working non-stop from the moment the exam papers were passed out. There was no way that any student could get the exam, then casually glance through their notes looking for solutions, because there just wasn't enough time. You had to understand immediately what each problem was asking and know the general approach to solving it. Then, you could turn to the exact right place in the textbook or in your notes to get the details on solving it. Plus, the exam problems usually included a twist such that they could not be solved in the exact same way as they were solved in the book. Instead, we would be forced to think carefully about each problem and find a means of solving it, making use of all available resources. I always thought this approach was much more revealing of the students' understanding of the subject, because it put the emphasis on problem solving, rather than on memorization.

I don't mean to say that there is no value in knowing rules and definitions, obviously there is. Rather, I believe this type of knowledge comes naturally with prolonged exposure to the subject.

13. May 9, 2005

### mathwonk

for many people it may not be important. but basically knowing the hypotheses of a theorem is like knowing the rules of the highway, it keeps you on the right side of the street.

i.e. theorems are not true except when the hypotheses hold. so knowing the hypotheses guides you in the use of the theorems.

if you always work in a restricted or elementary realm where there is no worry of whether they hold or not, then maybe you do not need to know them.

many people like to defend their lack of understanding of the foundations of a subject by belittling it as "memorization".

usually on a more comprehensive test than the sketch above I try to plumb the depth of understanding of theorems and their statements to expose the presence of memorization in the mindless sense.

for instance question 5 above, which is quite a useful statement follows if you understand the mean value theorem, but not if you have only memorized it thoughtlessly.

i took as my task here to examine someones knowledge of calculus, not merely of numerical applications of the techniques of calculus. one does not know what uses each student will put his knowledge to, so i expect them to learn as much as feasible.

Why limit what you learn just because you think you may not need it? You limit your own horizons in this way.

As Feynman states in his little introductory physics lecture series: [roughly]

"I am going to treat all of you here as if you were going to be physicists, even though that is only true for a small number of you. Indeed all your professors here at cal tech will treat you that way."

that is the standard of a high level school, and of a high level course.

moreover in mathematics there are usually a very small number of basic theorems which contain fundamental ideas that are used over and over. these are worth knowing by heart, tatooing them on your chromosomes as it were.

then these statements should of course be fleshed out by repeated applications and variations.

I myself have always been puzzled by the phenomenon of students who memorize statements and do not bother to reflect on their meaning.

just because it takes years to do this successfully is no argument not to begin early, and to keep at it. It has taken me personally years to understand the riemann roch theorem, my favorite theorem, well enough to remember the statement accurately.

Once in an attempt to get my scholars to learn just one theorem, the fundamental theorem of calculus, I tested it 8 times during the quarter, and offered extra credit for any correct statement of it if i should ever forget to test it.

Finally after 8 tests, all but one student was stating it correctly, viz. "if f is a function which is continuous on an interval I containing a, then the indefinite integral of f from a to x is a differentiable function on I whose derivative equals f."

I.e. even though they had just declared that the indefinite integral of any continuous function f does provide an antiderivative for f, they nevertheless maintained that this particular continuous function has no antiderivative.

On another test I once asked my students to prove question 5 above, that the only functions satisfying f' = f were constant multiples of e^x, and announced the problem in advance. most got it right.

flushed with success, and trying to provide an easy "gimme" on the final, I asked it again, but without announcing that i would.

most students got it wrong and declared me to blame. their reasoning was that when i announced it in advance they naturally memorized it without thinking about what they were doing, so when i asked it again, of course there was no residual knowledge.

It never dawned on me that anyone could be this stupid. students who are so foolish as to consciously decline to think about material they are told is of importance do puzzle and challenge me, i admit.

Perhaps in mathematics today students do need to be told that one does not ever memorize a theorem mindlessly, one has to think about it too. it is hard however to realize that such an obvious truth needs pointing out.

The whole idea of differential calculus is that of local approximation of a non linear function by a linear one.

The mean value theorem tells you how much information that linear approximation yields about the orginal non linear function.

Because I myself think hard about all the results i present each year, never satisfied to repeat the same presentation as found in books,l have occasionally been able to improve on them. e.g. i recently noticed that the rolle theorem together with the intermediate value theorem, implies that on an interval where a function's derivative is never zero, that function must be strictly increasing or strictly decreasing.

I call this the fundamental principle of graphing: on any interval where there are no critical points, a function is strictly increasing or decreasing."

It is hard to find this simple principle in any standard book, especially without appealing to the mean value theorem.

I noticed it only because I know the statement of the theorem well, and also pondered the proof for years. How could I do that if i did not even know the statement, and just looked it up in the book every time?

I also stress the power of logic in my courses, since to most of them I believe the ability reason will be mroe useful than the ability to apply techniques of calculus. hence statements as well as proofs of theoprems are a training ground for logical reasoning.

I point out that a successful appeal for a tax break is a theorem, hypothesis: whatever is given in the tax code; conclusion: i deserve that tax break. you cannot write a successful appeal if you do not know the statement precisely.

it is always a challenge to figure out what to ask a student to do, in order to maximize the usefulness of the knowledge he will carry away, but consciously minimizing it hardly seems a useful strategy.

after all some of the students may wish to be instructors, or may find them selves locked in the chateau d'if without refernce material and only their memories to sustain them.

Last edited: May 9, 2005
14. May 9, 2005

### mathwonk

its sort of like teaching a man to fish as opposed to giving him one meal. definitions and theorems contain ideas that have infinite applications.

someone who knows the definition of a derivative, has the potential of discovering for himself the derivative of any function, and is not dependent on the few he has memorized.

he is also able to verify the accuracy of the specific derivative he has in his memory.

just think about what a definition actually is: it is a precise codifying of a concept that has been found to be especially important, sometimes fundamentally so.

if a definition is a good one, NOT to learn it by heart should be considered especially foolish.

similarly a good theorem, like FTC, is a precise statement of one of the most important tools in a subject.

who knows more calculus: someone who knows the derivatives of x^2 and of sin(x), or someone who knows the FTC?

or what if someone can explain why the derivative of x^2 is 2x, or can explain what the FTC means?

Last edited: May 9, 2005
15. May 9, 2005

### mathwonk

let me try to distinguish between thoughtful memorizing and stupid memorizing.
some students try to memorize the volume of a cone as (1/3)pi r^2 h,
all strung together, every letter meaningless.

then they may forget the (1/3), or leave off the power.

this is stupid memorizing.

but if one remembers in general that volume is roughly = height times area of base,
at least for solids with vertical sides, then one can mentally separate out the parts of this formula as follows: pi r^2 is the area of the base, and h is the height, so
(pi r^2) (h) would be the volume of a cylinder,

but a cone has much less volume than a cylinder, even less than half, so probably it is (1/3(pi r^2 h.

this is more intelligent memorizing, i.e. partly memorizing, partly reasoning.
no one with this point of view would ever say the volume of a cone = pi r^2 h.

and if one knows that volume is measured in cubic units, i.e. is three dimensional, no one would ever mistake the volume formula of a cone, for the area formula, as often occurs.

by the way i myself made the mistake of leaving off the (1/3) from the volume formula of a cone on my SAT achievement test. I never learned to think intelligently about these things until I got out of high school.

16. May 9, 2005

### Poop-Loops

I still say the best kind of memorization is just looking up the formula in a book. :p

PL

17. May 10, 2005

### kdinser

Mathwonk, there is no question that what your saying is correct. However, going by the pace that my calc II class went, gaining that level of understanding of all the theorems would have been nearly impossible for all but the brightest students.

I truly loved the calc I class I took about 5 years ago and the college algebra class before that. Tons of graphs, plenty of real world problems and thought projects. I really felt like I came out of that class with a good understanding of basic derivative calculus and the very basics of integrals.

Then, 5 years after that, I took calc II. Looking back at my notes from that class, I can count about 12 graphs that I copied down all semester long. The pace of that class was absolutely brutal. I was taking 14 credits, a light load for an engineering student, and just had time to memorize the formulas before each test.

On the test dealing with infinite series, I didn't have the slightest clue why any of those convergent/divergent tests worked or were valid and neither did most of the class. I consider that pretty sad. I hope to have the time some day to go back and retake calc 1-3 just to increase my own understanding of the material, but it will be after I'm done with college and when I have more time to devote to the subject.

Maybe calculus should be broken into 4 semesters, 3 credits each or maybe that should be an option for those of us that aren't as quick with mathematics as the math majors. I enjoyed the pace of my diff eq course this year. We seemed to spend more time understanding why the procedures used to solve differential equations worked then we did trying to memorize the procedure itself.

18. May 10, 2005

### gravenewworld

19. May 10, 2005

### mathwonk

you are missing my point. I am not telling you how much calc you should know, but how much you do know, with my test. and that is a non honors test.

there are not that many theorems to know actually in calculus:

here they all are:

1) definition of a derivative (one line)

2) rules for derivatives: easy except for leibniz product rule and if you know that the deriv of a poly at zero is the linear term you can always remember it by multiplying out (a+bx)(c+dx) = ac + [ad+bc]x + bdx^2, i.e. (fg)' = fg' + f'g.

3) chain rule: deriv of a composite is product of derivs: to remember use leibniz notation: dy/dt = (dy/dx) (dx/dt).

4) intermediate value theorem; if f(a) < 0 and f(b) > 0 and f cont between a and b, then f = 0 somewhere in between. obvious.

5) max - min value thm: a function cont at and between a and b, has a maximum somewhere at or between them.

cor: rolle theorem: if f is differentiable everywhere from a to b, and f(a) = f(b) then f' = 0 somewhere between a and b. draw a picture.

cor: on an interval where f' never =0, a diffble f cannot take the same value twice, hence f must be strictly monotone.

cor: mean value theorem: if f is diffble between a and b then the secant line joining (a,f(a)) and (b,f(b)) has same slope as some tangent line in between.

cor: if the slope of every tangnet line is zero, then the slope of every secant line is zero, hence f is constant if the deriv is always zero.

cor: if f">0 everywhere then f is increasing.

cor: if f'=g' everywhere then f-g is constant.

Thats it. all of diff calc in a nutshell.

anyone who cannot memorize that much in 15 weeks is in serious difficulty in college. (not even one sentence per week)

now i submit that it is useless to memorize it without trying to understand it. what one should try to commit to memory is this sequence of ideas, possibly in ones own words, but it should eventually be possible to remember all the salient facts, and explain what they mean, and use them.

e.g. why do some of the statements say "cor"? and how is this justified?

All we are trying to do here is understand the meaning of slope of a tangent line to a graph, and then know how that slope informs us about the original graph. is that too much to remember?

most students i have queried do not even know that a derivative is the slope of a tangent line, a few months after the course. i wonder what they did learn?

If you do not remember anything else, remember that diff calc is about approximating curves by straight lines. then think about that.

Last edited: May 10, 2005
20. May 10, 2005

### mathwonk

the goal of learning a subject should be to boil it all down to a few essential ideas, and reflect deeply on those. then one can regenerate all the rest from them when needed.

calculus contains only three concepts: continuity, differentiability, and integrability. think about that.

you need to know the definitions of these concepts, and their main properties ("theorems").

then learn how to use them, i.e. what they can tell you, and how you can use this knowledge. thats all.

it takes a lot of study time to master the material this well, but it is worth it, as then you never lose it. there are some students who actually do this, by the way. they are the ones who are not studying just before a test, but are helping others.

Last edited: May 10, 2005