Pianostreet calculus group

[rachmaninoff_forever]

Bob decides to buy some ice cream.  The ice cream cone has a radius of 10cm, and a height of 20cm.  Before Bob begins to eat the ice cream, he wants to solve his Rubik’s cube first.  While he’s solving it, the ice cream is melting through the bottom of the cone at a rate of 3 cubic centimeters per minute.  When he finally solve the cube, the depth of the ice cream is only 11cm now.

What is the rate of the ice cream melting when the depth of the ice cream is at 11cm?


Do you guys think you could do this problem for me please?  We're doing a project, and I wanna make sure I get this right.

[lloyd_cdb]

Umm, Aren't you given the answer in the problem?  If it's melting through the bottom of the cone at a rate of 3 cubic centimeters per minute, it shouldn't change the rate unless you're looking for a % change*.


*assuming laws of thermodynamics, fluid dynamics etc. don't apply. <--- I don't know any of that, just saying.

[rachmaninoff_forever]

Umm, Aren't you given the answer in the problem?  If it's melting through the bottom of the cone at a rate of 3 cubic centimeters per minute, it shouldn't change the rate unless you're looking for a % change*.


*assuming laws of thermodynamics, fluid dynamics etc. don't apply. <--- I don't know any of that, just saying.

I'm doing my homework.

And...

And...

And...

I'm...

Enjoying it?

[lloyd_cdb]

I'm doing my homework.

And...

And...

And...

I'm...

Enjoying it?

Just like Bach, you eventually warm to it.

[rachmaninoff_forever]

Just like Bach, you eventually warm to it.

Hey Bach IS NOT the same thing as math!  I would do math over Bach any day!

[starstruck5]

I would have thought that by now anyone could use computers to solve any mathematical problem -so that, those who hate math would never need to actually learn anything but the fundamentals -and how to master software -like you master photoshop -

It seems our educationalists are still living in the 19th century - get with the programme you guys -

[lloyd_cdb]

I hated english, history, chemistry, biology and foreign languages.  Let's just not teach them because a few people don't like it.  Basic math is everywhere.  Pretending that it doesn't exist just because someone or something else will solve it is ridiculous.  How do you expect math students to be interested in it and develop this software if they don't study it in the first place?  That applies to any subject.  If I had never bothered learning to read because there are programs that dictate it to me, would that be the same concept?  People are afraid of math because they think only a few people can understand it.  It's just like language or music, people can understand the basics it if they put in the effort.  That doesn't mean everyone will be a mathematician, just like not everyone is a concert pianist or famous novelist.  However, not teaching math at a elementary and high school level would be as disastrous to society as not bothering to teach people to read.

[lloyd_cdb]

To add to that, from the National Center for Education Statistics:

Question:
What are the most popular majors for postsecondary students?

Response:
Of the 1,650,000 bachelor's degrees conferred in 2009–10, the greatest numbers of degrees were conferred in the fields of business (358,000); social sciences and history (173,000); health professions and related programs (130,000); and education (101,000). At the master's degree level, the greatest numbers of degrees were conferred in the fields of education (182,000) and business (178,000). At the doctor's degree level, the greatest number of degrees were conferred in the fields of health professions and related programs (57,700); legal professions and studies (44,600); education (9,200); engineering (7,700); biological and biomedical sciences (7,700); psychology (5,500); and physical sciences and science technologies (5,100)

All of the bolded sectors have a large reliance on mathematics.

Computer science is one of the fastest growing sectors.  Have you ever learned programming? There is TONS of math involved.  I just don't see how you could even think of eliminating a sector of education because "someone else will do it for me".

[ajspiano]

I would have thought that by now anyone could use computers to solve any mathematical problem -so that, those who hate math would never need to actually learn anything but the fundamentals -and how to master software -like you master photoshop -

It seems our educationalists are still living in the 19th century - get with the programme you guys -

 

you're trolling us right? what exactly do you class as the fundamentals?

[lloyd_cdb]

you're trolling us right?

I bit

[littletune]

Well I haven't been reading this thread at all so I'm not sure if it's ok that I ask this here but I was just wondering: how much is  :
∞ - 1 ?
Can you get a real number that way? or could a computer get it? or not really? 

[lloyd_cdb]

lol, thats a fun question.  There are so many fun (and wrong) proofs you can give to your calc teacher in high school to confuse the crap out them.  In less than technical terms:

∞ - 1 = ∞, so if you subract ∞ from each side, aren't you left with 0 = 1?!?!?!? OMG THE WORLD IS GONNA END, RUN!!!!!!!

In technical terms, ∞ isn't really a number.  It's called a limit.  It might be incredibly confusing to put into words without symbols, but I'll give it a shot.

Using summation, you can sum the number x from 0 towards infinity. I realize this reuses the "number" which can make it confusing, but you aren't actually ever getting to that number.  So in the simple case:

summation of x from n to infinity = 0 + 1 + 2 + 3 + 4 + 5 + 6 + ...n.  Clearly you can actually calculate any of these solutions for any number.  But the LIMIT as the number n gets larger is infinity.  Summation notation and limits can be extremely useful in higher level math.  I haven't really used much of this in the last 6 years, so I can't really give all that much more of an explanation without digging into my books and reviewing for a bit.

[starstruck5]

Nope -I mean it -computers can bring maths alive in a way the pre-computer age never could -

The fundamentals is a familiarity with basic functions -you would know what a graph was -what x and y represented -and later Z  Also the Trig stuff -It would also depend to a large degree on what the software was capable of -and could teach -I am no maths expert or educator -Graphs and functions can be beautiful -it is just when they are presented in a dusty way, it makes it seem boring as hell --I actually fell asleep in a maths lesson many years ago now -and the problem given by Rach-forever was sooooo dull -it made me think who actually cares -come on guys -make maths sing!  Only computers can do that for the mathematically challenged -

[chadbrochill17]

Here's another fun idea with infinity.

Start at zero and then go to positive infinity. Then, go from negative infinity to positive infinity.
Which is bigger?

[lloyd_cdb]

starstruck,

Of course computers can bring math alive beyond slide-rules, punch cards, calculators, and the rest of the evolution of technology.  The problem is that these mathematicians need to come from somewhere.  There is so much more math to figure out in the universe beyond just calculating pi to 15 billion decimal places.  Much of math comes from simplification of higher math into basic elements.  Knowing much of the simpler aspects can actually give you a fighting chance of solving incredibly complex problems.  Computers become a crutch for the mathematically challenged.  By the way, I've fallen asleep in math class as well, but I don't believe thats a function of the elements of the class, but more in regards to the teaching method.  

I personally enjoy the dull problems because I can't learn from people instructing me how to do things.  In my college math classes, I went to MAYBE 25% of them.  Before each semester, I would talk to my professors and let them know I just couldn't learn in the standard educational fashion.  Instead of going to class, I would just work through EVERY problem in the textbook and ask for additional problems I could do.  That is the only way I learn.  My talents are in deductive reasoning.  I can't necessarily remember every formula, but I can derive them from basic mathematical common sense.  This is why I bombed almost every single one of my 'memorization classes'.

This is the exact problem with education.  It's not in teaching things that could be solved in easier fashion.  It's in teaching everyone the exact same way.  Just as in piano, some people learn by ear, others read, some people memorize through physical memory.  Some are better at improv because they might just understand music theory more than they can play someone elses music that's all but set in stone.

Lastly, higher level math doesn't even use numbers.  I stopped using numbers in my classes after sophomore year.  A computer can't solve anything I learned after sophomore year.  It needs to be figured out by people so that there are actual formulas to apply in the programs that run your numbers.  How do you think derivatives were understood?  Proofs with no numbers that can then be applied to numbers.  How about statistics?  Proofs are what figured out normal distributions, students-t distribution, chi^2 distributions, etc.  All these numbers are what plot that graph in the background when you give a table of numbers.  Saying that computers can do everything for you is just like saying 'everything that can be invented has been'.

EDIT:

https://www.aomori-u.ac.jp/staff/midori/ProbDistr/t-e.pdf

Try wrapping your head around that one without learning any of the derivations of mathematical formulas.

BTW, that's the dumbed down formula.

Feel free to take a look at this powerpoint presentation.

[starstruck5]

starstruck,

Of course computers can bring math alive beyond slide-rules, punch cards, calculators, and the rest of the evolution of technology.  The problem is that these mathematicians need to come from somewhere.  There is so much more math to figure out in the universe beyond just calculating pi to 15 billion decimal places.  Much of math comes from simplification of higher math into basic elements.  Knowing much of the simpler aspects can actually give you a fighting chance of solving incredibly complex problems.  Computers become a crutch for the mathematically challenged.  By the way, I've fallen asleep in math class as well, but I don't believe thats a function of the elements of the class, but more in regards to the teaching method.  

I personally enjoy the dull problems because I can't learn from people instructing me how to do things.  In my college math classes, I went to MAYBE 25% of them.  Before each semester, I would talk to my professors and let them know I just couldn't learn in the standard educational fashion.  Instead of going to class, I would just work through EVERY problem in the textbook and ask for additional problems I could do.  That is the only way I learn.  My talents are in deductive reasoning.  I can't necessarily remember every formula, but I can derive them from basic mathematical common sense.  This is why I bombed almost every single one of my 'memorization classes'.

This is the exact problem with education.  It's not in teaching things that could be solved in easier fashion.  It's in teaching everyone the exact same way.  Just as in piano, some people learn by ear, others read, some people memorize through physical memory.  Some are better at improv because they might just understand music theory more than they can play someone elses music that's all but set in stone.

Lastly, higher level math doesn't even use numbers.  I stopped using numbers in my classes after sophomore year.  A computer can't solve anything I learned after sophomore year.  It needs to be figured out by people so that there are actual formulas to apply in the programs that run your numbers.  How do you think derivatives were understood?  Proofs with no numbers that can then be applied to numbers.  How about statistics?  Proofs are what figured out normal distributions, students-t distribution, chi^2 distributions, etc.  All these numbers are what plot that graph in the background when you give a table of numbers.  Saying that computers can do everything for you is just like saying 'everything that can be invented has been'.

EDIT:

https://www.aomori-u.ac.jp/staff/midori/ProbDistr/t-e.pdf

Try wrapping your head around that one without learning any of the derivations of mathematical formulas.

BTW, that's the dumbed down formula.

Feel free to take a look at this powerpoint presentation.

Thanks for your considered and passionate response!  I know mathematics doesn't exist just to torture those who can't get their head around it -it just seems that way.

[lloyd_cdb]

Believe me, I agree it can be INCREDIBLY dull at times and I realize not everyone will be interested in it.  I just think the option should always be given.

I'm sure we'd have some serious essays on this forum if someone showed up and said "we shouldn't teach music theory. It all just comes from the radio." Not the best comparison, but I giggled when thinking about it.

[chadbrochill17]

There also exists formulas that CAN NOT be solved WITHOUT a computer. Some things need to be solved numerically, like a lot of cosmology, so it does go both ways.

[starstruck5]

Wish I could work out a mathematical way to win the lottery!  It is all a bit too random though and the odds against finding a winning line are ridiculous -

[mikeowski]

Wish I could work out a mathematical way to win the lottery!  It is all a bit too random though and the odds against finding a winning line are ridiculous -

Something like a winning line doesn't exist. Any random combination has the same chance as the next.

[lloyd_cdb]

Something like a winning line doesn't exist. Any random combination has the same chance as the next.

On the other hand, there are winning numbers that are less likely to be split with other people.  Since people who pick numbers tend to pick less randomly, they end up picking numbers that are familiar to them such as birthdays, anniversaries, the time their child was born etc. And knowing that there are only 12 months in a year, 30 days in each month, 24 hours in a day... etc., the distribution of picks tends to be crowded in the lower numbers.  That being said, if the winning numbers are in the lower end of the spectrum, people end up splitting it more frequently.  If they are in the higher spectrum, there is a higher probability that there will be a sole winner.  Again though, as you said, it's a uniform distribution on the actual winning numbers.  But since that's the case, picking numbers in the higher spectrum actually give you a higher expected value while giving you an equal chance of winning.

[starstruck5]

On the other hand, there are winning numbers that are less likely to be split with other people.  Since people who pick numbers tend to pick less randomly, they end up picking numbers that are familiar to them such as birthdays, anniversaries, the time their child was born etc. And knowing that there are only 12 months in a year, 30 days in each month, 24 hours in a day... etc., the distribution of picks tends to be crowded in the lower numbers.  That being said, if the winning numbers are in the lower end of the spectrum, people end up splitting it more frequently.  If they are in the higher spectrum, there is a higher probability that there will be a sole winner.  Again though, as you said, it's a uniform distribution on the actual winning numbers.  But since that's the case, picking numbers in the higher spectrum actually give you a higher expected value while giving you an equal chance of winning.

Thanks for that insight -very interesting.

I know mathematicians say that every set of winning numbers is equally likely -but I have noticed that there is never a set which has perfect numbers -by that I mean perfect squares -such as 1 4 9 16 25 36 -or a perfect Fibonacci sequence -1-4-7-11-18-29 -When I analysed past numbers a few months back, this was true -I did find some sets which had some perfection -but always with a spoiling number -

I realise that there are many more imperfect sets than perfect ones- but I have yet to see any sequence of 6 consecutive numbers -obviously I can't help thinking then, that some sets are less likely than others -

One other point -when I played Poker I was fortunate enough to be dealt a Royal Flush -what are the odds -do you know?  Someone told me that I would never win the lottery, because you would never overcome those kind of odds twice -Then again, being born must be much higher than winning any lottery -something like winning the lottery ten times in a row! Even higher maybe -

[chopin2015]

OMG OMG OMG I JUST FOUND OUT ABOUT THE EXISTENCE OF *strange synthesizer music plays* Stochastic anything-calculus and oscillators. WT why was I born yesterday?

[perprocrastinate]

gg calc exam. I got a 96.

I nearly broke my spine the two nights I studied for it. Totally worth it.

Now I can bask in my brief moment of glory before I bomb the next test.

[rachmaninoff_forever]

gg calc exam. I got a 96.

I nearly broke my spine the two nights I studied for it. Totally worth it.

Now I can bask in my brief moment of glory before I bomb the next test.

I'm still failing almost every test.

I've come to the conclusion that I'm just a bad test taker.

[rachmaninoff_forever]

What's the derivative of cos(3x)?

[chadbrochill17]

What's the derivative of cos(3x)?

-3*sin(3x).

[rachmaninoff_forever]

-3*sin(3x).

Where did the -3 come from?

[austinarg]

Where did the -3 come from?

The derivative of cos(x) is -sin(x).

[chadbrochill17]

Where did the -3 come from?

remember to do the chain rule. You must multiply the derivative of the function by the derivative of whatever is inside the function. The derivative of cos is -sin, while the derivative of 3x is 3.

[littletune]

lol, thats a fun question.  There are so many fun (and wrong) proofs you can give to your calc teacher in high school to confuse the crap out them.  In less than technical terms:

∞ - 1 = ∞, so if you subract ∞ from each side, aren't you left with 0 = 1?!?!?!? OMG THE WORLD IS GONNA END, RUN!!!!!!!

In technical terms, ∞ isn't really a number.  It's called a limit.  It might be incredibly confusing to put into words without symbols, but I'll give it a shot.

Using summation, you can sum the number x from 0 towards infinity. I realize this reuses the "number" which can make it confusing, but you aren't actually ever getting to that number.  So in the simple case:

summation of x from n to infinity = 0 + 1 + 2 + 3 + 4 + 5 + 6 + ...n.  Clearly you can actually calculate any of these solutions for any number.  But the LIMIT as the number n gets larger is infinity.  Summation notation and limits can be extremely useful in higher level math.  I haven't really used much of this in the last 6 years, so I can't really give all that much more of an explanation without digging into my books and reviewing for a bit.

Thanks for answering!  but what is infinity then?  I mean is it anything at all? or people just say infinity when they don't know what else to say and when they can't imagine numbers anymore? I mean I know it's a limit, but why is there a limit that's not really a limit cause it's infinite? cause if that's the way it is then it's kinda like: oh this is the way it is because it's the way it is...  doesn't sound very scientific or mathematical to me.  I mean it seems to me that the only way something could be infinite and would stay the same no matter what you would take from it (or add) is if it was a circle or something like that or if it was getting larger all the time! (but how and for how long it would be getting larger, for infinity?) Other than that it seems to me like people only invented it cause it gets too difficult to understand! Just like when people don't know why some things are the way they are or how they started to exist and they just say: oh a higher power created them that way, cause a higher power can do anything. And I'm not saying there's no higer power but it still doesn't answer the real question.

[austinarg]

Thanks for answering!   but what is infinity then?  I mean is it anything at all? or people just say infinity when they don't know what else to say and when they can't imagine numbers anymore? I mean I know it's a limit, but why is there a limit that's not really a limit cause it's infinite? cause if that's the way it is then it's kinda like: oh this is the way it is because it's the way it is...   doesn't sound very scientific or mathematical to me.   I mean it seems to me that the only way something could be infinite and would stay the same no matter what you would take from it (or add) is if it was a circle or something like that or if it was getting larger all the time! (but how and for how long it would be getting larger, for infinity?) Other than that it seems to me like people only invented it cause it gets too difficult to understand! Just like when people don't know why some things are the way they are or how they started to exist and they just say: oh a higher power created them that way, cause a higher power can do anything. And I'm not saying there's no higer power but it still doesn't answer the real question.

Well, infinity is anything but a number, I can tell you that. Imagine if someone asked you "Tell me the closest number to 6". You could say 5.99, 5.999, 5.9999, but you would always find a bigger number. Because there is no "biggest number", real numbers are an infinite set.

[chadbrochill17]

Thanks for answering!  but what is infinity then?  I mean is it anything at all? or people just say infinity when they don't know what else to say and when they can't imagine numbers anymore? I mean I know it's a limit, but why is there a limit that's not really a limit cause it's infinite? cause if that's the way it is then it's kinda like: oh this is the way it is because it's the way it is...  doesn't sound very scientific or mathematical to me.  I mean it seems to me that the only way something could be infinite and would stay the same no matter what you would take from it (or add) is if it was a circle or something like that or if it was getting larger all the time! (but how and for how long it would be getting larger, for infinity?) Other than that it seems to me like people only invented it cause it gets too difficult to understand! Just like when people don't know why some things are the way they are or how they started to exist and they just say: oh a higher power created them that way, cause a higher power can do anything. And I'm not saying there's no higer power but it still doesn't answer the real question.

So in a sense you are right; there is no such thing as infinity. However, in math infinity is the essence of calculus, as we take limits going to infinity, summing up infinitesimally small intervals and the like. You can't think of it as a number because it isn't. You can think of it as, "What would happen if we made this interval as small as we can physically imagine, and then we made it even smaller?" Or you can also think of how 0.999999999... is the same as 1, given an infinite number of decimals.

[rachmaninoff_forever]

What's the antiderivative of 8/³√x ?

[patrickd]

What's the antiderivative of 8/³√x ?

I haven's used integrals in some time, but I got 12x^(2/3)+c, so correct me if I am wrong.

[lloyd_cdb]

I got the same:

Rach,

when you ask, let us know your answer and how you did it so we can actually help instead of doing your HW

integration by parts formula: ∫udv = uv -  ∫vdu

 8/³√x = 8 * x^^-1/3

I tend to make the simple aspect u, so du is even simpler.

u = 8, du = 0
dv =  x^^-1/3 dx, (integrate) v = 3/2x^^2/3

plug those into the formula,

∫8/³√x = 8(3/2x^^2/3) - ∫(3/2x^^2/3)(0)

∫8/³√x = 8(3/2x^^2/3) = 12x^^2/3 + c

[austinarg]

It is not necessary to integrate by parts, 8 is a constant value. Just integrate x^-1/3.

[rachmaninoff_forever]

I just wanna do enough in this class so I won't get rejected in the middle of July.

I'll be happy with a D.

#second semester senior

[iansinclair]

Just reading this thread makes my head ache.

I knew there was a reason I dropped out of any further math courses after freshman year in college...

[lloyd_cdb]

It is not necessary to integrate by parts, 8 is a constant value. Just integrate x^-1/3.

Yep, dumb of me. On the other hand, I'm impressed I remembered the formula lol.

[rachmaninoff_forever]

Find the average value or csc^2x over the interval from x = pi/6 to x = pi/4 please.

[lloyd_cdb]

avg(f(x)) from a to b = (_a∫^bf(x)dx) / (b-a) = [F(b) - F(a)]/(b-a)

∫csc(x)^2 = -cot(x)

so it's [-cot(pi/4) - (-cot(pi/6))] / (pi/12)

I think. No guarantees, this is stuff I don't remember.

[chopin2015]

So jelous of people who are good at math! Keep practicing it, yall!

[slobone]

Perhaps this video will explain it a bit more clearly...

[rachmaninoff_forever]

So jelous of people who are good at math! Keep practicing it, yall!

I feel you I am so jealous of the kids who are actually doing decent in class. 

Anyways, for our last take home test, a bunch of people got caught for copying the answers from the previous AP test key online.  I cheated too but didn't get caught!

Those idiots were freaking copying the answers verbatim.

[ajspiano]

Those idiots were freaking copying the answers verbatim.

Exercising your intelligence in this regard will probably serve you better in life than knowing the answers in your own right anyway.

[lloyd_cdb]

Exercising your intelligence in this regard will probably serve you better in life than knowing the answers in your own right anyway.

Life is all about the game, not the knowledge.

[perprocrastinate]

I feel you I am so jealous of the kids who are actually doing decent in class.

Funny story about doing "decently". When two sophomores in my class got 100s on the recent test, they made the entire class feel stupid.

EDIT: Ah, a perfect opportunity to use a phrase you coined on this forum. The rest of the class (including me) felt sooooo salty!

[rachmaninoff_forever]

Funny story about doing "decently". When two sophomores in my class got 100s on the recent test, they made the entire class feel stupid.

Gah I hate those people!  They're freaking monsters!

Or the foreign exchange students, they're animals!

I remember we were doing some problems in a group, and this is what went down...

Teacher:  okay guys, I want you to do this in your groups.

Me:  alright guys, I think I found the answer to number 1:

Foreign exchange student:  WHAT?!?!?!?!. I'm already on number 10!!!!

Me:  teacher, can I drop the class?

I felt sooo salty!!!

[chopin2015]

Gah I hate those people!  They're freaking monsters!

Or the foreign exchange students, they're animals!

I remember we were doing some problems in a group, and this is what went down...

Teacher:  okay guys, I want you to do this in your groups.

Me:  alright guys, I think I found the answer to number 1:

Foreign exchange student:  WHAT?!?!?!?!. I'm already on number 10!!!!

Me:  teacher, can I drop the class?

I felt sooo salty!!!

Lol i always start off ahead of the class but by the end im like sitting alone in the stupid seat which changes based on where i sit.