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[BlueMagic]

You have to follow the exact same procedure.

if if f'(x)= (x^2-1) / x^2

Then you
1)differenciante the numerator, multiply it the denominator, which gives you:

[1] 2x * x^2 = 2x^3

2)differenciate the denominator, multiply it the numerator, which gives you:

[2] 2x*(x^2 -1) = (2x^3 - 2x)

3)You do [1] - [2] and divide that by the denominator squared. Which gives you:

([1] - [2])/x^4 = (2x^3 - 2x^3 +2x) /x^4 = 2x/x^4 = 2/x^3

 

[ZeroRay]

Thanks I'll try that.

 

[vordhosbn]

3 friends dine in a fancy restaurant. After the meal the waiter brings the bill - all together 50$. The friends pay these 50$ and the waiter brings the 50$ to his boss. The boss told the waiter: "Hey, these 3 gentlemen are regulars, and regulars get a 10% discount! Here are 5$ give them back!"
On his way back to the table of the friends, the waiter thinks by himself: "These scroungers haven't given me any tip, so I'll keep 2$ and return them only 3$."

Okay, at the end the 3 friends have payed 47$ for their meals, right? (50-3=47)
2$ has the waiter now in his own pocket.
But wait, 47+2=49 !(?) Suddenly there is 1$ missing! Where is it ???

Anyone? This is pretty much recreational, I just need to know wtf is missing.

 

[whatsinaname]

Anyone? This is pretty much recreational, I just need to know wtf is missing.

Hmm, they paid 50$ and got back 3$ (which is in their pocket). So they paid 47$.

But the waiter gave 45$ to the owner. So 47$ - 45$ = 2$.

 

[Prophet Steve]

Yeah, what he says, you have to do minus and not plus.

50 dollars paid, they had to pay 45, he keeps two and gives three back.
So they got three of the difference and he got the other two.

EDIT:

Maybe this makes it easier.
It is right that they have paid 47. 45 for the food and 2 for the guy. The other 3 they got back.

They gave 50, minus three that they got back, so they payed 47. They should have paid 45, and he got the other two.

Well, one of this explanations must make it clear to you.

 

[RedShift]

Anyone? This is pretty much recreational, I just need to know wtf is missing.

Well for a start you say they paid $47 for their meal, then count the $2 the waiter kept again, when that is included in the $47 they paid, so in fact there is only $47 at the end. But you also forgot to count the $3 they got given as change. So that adds up to the original $50 from the start.

Its a stupid riddle based on using confusing language and information overload.

'Proofs' that 2=1 etc. by dividing by zero are equally annoying. Mostly because as someone who's going to do a maths degree I get loads of smug people using them to show me that actually maths doesn't work, and I'm wasting my time

 

[Lagspike_exe]

Hey, GAF, I need help with this problem:

1 + sin2ALPHA
--------------- = SQUARE ROOT 2 * (90 - ALPHA)
sinALPHA*cosALPHA

 

[malsumis]

Hey, GAF, I need help with this problem:

1 + sin2ALPHA
--------------- = SQUARE ROOT 2 * (90 - ALPHA)
sinALPHA*cosALPHA

what do you want to do with this?

 

[vordhosbn]

Just wanted to say you guys are awesome, excuse the late reply this thread was hard to find.

I expect the math majors to know this one, but what's so special about this number?

142857

EDIT: got it, it's a cyclical number.

 

[Anth0ny]

omg im dying here:

The net monthly profit, in dollars, from the sale of a certain item is given by the formula P(x)= 10^6 [ 1+ (x - 1) e^-0.001x ], where x is the number of items sold. Determine the number of items that yield the maximum profit. At full capacity, the factory can produce 2000 items per month.

I'm using power rule, but I think I'm deriving the inside of the bracket wrong. Thanks in advance!

 

[close to the edge]

omg im dying here:

The net monthly profit, in dollars, from the sale of a certain item is given by the formula P(x)= 10^6 [ 1+ (x - 1) e^-0.001x ], where x is the number of items sold. Determine the number of items that yield the maximum profit. At full capacity, the factory can produce 2000 items per month.

I'm using power rule, but I think I'm deriving the inside of the bracket wrong. Thanks in advance!

Does this help?
http://www.wolframalpha.com/input/?i=derive+10^6+(+1++(x+-+1)+e^(-0.001x))
You need to click on "show steps".

 

[Anth0ny]

Does this help?
http://www.wolframalpha.com/input/?i=derive+10^6+(+1++(x+-+1)+e^(-0.001x))
You need to click on "show steps".

Wow greatest website ever. Thanks a lot!

 

[close to the edge]

Wow greatest website ever. Thanks a lot!

Yup, Wolfram Alpha is pretty kickass for every "exact science".

 

[shadowsdarknes]

Hi, If anyone would like to help me, I've got a relatively simple question.

I'm taking Calculus this summer and I haven't taken pre calculus and I'm at somewhat of a disadvantage from the other students who have taken it or are taking calculus over again as a refresher.

I'm learning derivatives and I've got a problem where f(x)=e^2x
My book says f'(x)=2e^2x.
Can someone explain to me why this is?

 

[Rich Uncle Skeleton]

I'm learning derivatives and I've got a problem where f(x)=e^2x
My book says f'(x)=2e^2x.
Can someone explain to me why this is?

You can show this using either the product rule or the chain rule, the latter probably being the easier of the two. If you compose the function g(x) = e^x with the function h(x) = 2x you get g(h(x)) = e^(2x) = f(x), so the chain rule says:

f'(x) = g'(h(x)) h'(x) = e^(h(x)) (2) = 2e^(2x)

 

[Artanisix]

For calculus people:

Find the equation of the tangent line at x = 2 given f(0)=1 and f'(0)=-3

How do I go about solving this?

 

[close to the edge]

For calculus people:

Find the equation of the tangent line at x = 2 given f(0)=1 and f'(0)=-3

How do I go about solving this?

I haven't done much calculus yet but my best guess is using this definition:
http://en.wikipedia.org/wiki/Derivative#Definition_via_difference_quotients

Hope it helps.

 

[Therion]

For calculus people:

Find the equation of the tangent line at x = 2 given f(0)=1 and f'(0)=-3

How do I go about solving this?

I don't think that's enough information to come up with a unique answer. Do you have any more on f? "f is a polynomial of degree n" or something similar?

 

[Artanisix]

I don't think that's enough information to come up with a unique answer. Do you have any more on f? "f is a polynomial of degree n" or something similar?

No, that was it. It's a sample problem my professor wrote down on the board to prepare us for a test later tonight, and it's the only question he gave us that I haven't been able to really answer. Maybe he didn't write it all down.

 

[deadbeef]

For calculus people:

Find the equation of the tangent line at x = 2 given f(0)=1 and f'(0)=-3

How do I go about solving this?

How do you even know f is defined at x=2?

 

[Therion]

No, that was it. It's a sample problem my professor wrote down on the board to prepare us for a test later tonight, and it's the only question he gave us that I haven't been able to really answer. Maybe he didn't write it all down.

Well, I wouldn't feel bad about being unable to answer it then, 'cause as stated there would be infinite solutions.

Or else I missed something when I read it, which isn't impossible.

How do you even know f is defined at x=2?

Or no solution at all, in this case.

 

[Rich Uncle Skeleton]

I'm assuming it's a typo and either the 2 is supposed to be a 0 or both 0s are supposed to be 2s. In which case f gives you a point on the line, (0,1) or (2,1) depending on where was assume the typo(s) is/are, and f' gives you the slope, -3. From there you can easily write down the equation for the line in point-slope form.

 

[barnone]

Just started up Calc 2 today as a freshman (used AP credit to skip calc 1 at college) and had my first lecture on sequences. However, I am a bit confused about "epsilon". Can someone explain to me what

1. "epsilon > 0" means? (Seen under "Precise Definition of Limit" here)

2. What "n epsilon N" means (Seen here on wikipedia)

Here I see that an epsilon is a positive infinitesimal, so wouldn't it always be greater than 0 by definition?

 

[zoku88]

Just started up Calc 2 today as a freshman (used AP credit to skip calc 1 at college) and had my first lecture on sequences. However, I am a bit confused about "epsilon". Can someone explain to me what

1. "epsilon > 0" means? (Seen under "Precise Definition of Limit" here)

2. What "n epsilon N" means (Seen here on wikipedia)

Here I see that an epsilon is a positive infinitesimal, so wouldn't it always be greater than 0 by definition?

For 1. When they say epsilon > 0, what epsilon is is undefined (though, it's always used as a positive infinitesimal.) It's just so that they can say "for some epsilon > 0, f(x) - e = 0, blah blah" ie, so that you don't use symbols that aren't defined.

It's just a formal thing. It's not wrong not so say "for some eps > 0", it just wouldn't be well-written.

2. I don't remember my set symbols very well, but I believe that should mean, all x_n s.t. n is a member of the set N (where N is the set of all natural numbers?)

 

[Mudkips]

Just started up Calc 2 today as a freshman (used AP credit to skip calc 1 at college) and had my first lecture on sequences. However, I am a bit confused about "epsilon". Can someone explain to me what

1. "epsilon > 0" means? (Seen under "Precise Definition of Limit" here)

2. What "n epsilon N" means (Seen here on wikipedia)

Here I see that an epsilon is a positive infinitesimal, so wouldn't it always be greater than 0 by definition?

Copied from your link:

1. We say that LIM [n -> inf] a_n = L if for every number eps > 0 there is an integer N such that |a_n - L| < eps whenever n > N

This just means that the limit exists (and is equal to L) only if there is indeed a limit.
It means that for every positive number (epsilon), there must be a point in the sequence after which the terms are always less than that positive number away from the limit (in either direction).

For example, if you pick an arbitrary epsilon of 500, that just means that terms N+1, N+2, ... all have to be within 500 of the limit. (Never getting further away from the limit than epsilon after some point N).

But a boundary of 500 (in both directions) doesn't do shit to help us define a limit. Thus, the theorem states "for EVERY epsilon". This means that this has to hold for 0.000000001 just as it did for 500. The point in the sequence past which this is true will be very far out, typically, but that doesn't matter. What matters is that the sequence stabilizes at some point, and then always gets closer to (or stays the same distance from) the limit the further out you go.

Oh, and yes, "n E N" just means "such that n is a member of N". (This isn't the epsilon you're looking for.)

 

[Rich Uncle Skeleton]

Here I see that an epsilon is a positive infinitesimal, so wouldn't it always be greater than 0 by definition?

What other people said is correct; just wanted to address this part specifically: epsilon is not some fixed number like pi. It's merely a convention to use epsilon to represent small positive numbers, just as it's a convention to use x and y to represent arbitrary real numbers and n and m to represent integers. So you still need to specify that you're only considering positive real numbers epsilon. The definition just says that however close you want to get to the limit L (say, within epsilon = 1/2 or epsilon = 1/1,000,000), you can find a point in the sequence after which every term is that close.

Oh and that symbol you're talking about in 2 is not an epsilon. It's another symbol that, as others pointed out, means "is an element of". Here's a wikipedia link.

 

[barnone]

Hey thanks everyone, that helped me out plenty. If anyone knows of any online resources to complement Calc 2 studies that'd be fantastic (already using khanacademy and Paul's Online Notes as well as a plethora of on-campus assistance). The syllabus lists infinite sequences and series, volumes and work, techniques of integration, taylor series, polar coordinates, and differential equations as some of the topics. Again, thanks for the help, I'll probably check in here a few times throughout the year.

 

[AcciDante]

Can someone help me with this calc problem? It's more of an algebra question, though.

There's the whole problem from the solutions manual, but I'm not following how they got that numerator in the second step of the part I put a red box around.

 

[torontoml]

They made the top denominators the same by multiplying each side by what it didn't have.

Left side x sqrt[a + 2]

right side x sqrt[a + h + 2]

 

[AcciDante]

Thanks, duder. If there's radicals in fractions, my brain just shuts down :lol

 

[Scalemail Ted]

Hello,

quick math inquiry. I'm working on a population growth problem and I'm a bit unsure how to set this up to solve for a solution. Any advice?

 

[TimesEunuch]

Hello,

quick math inquiry. I'm working on a population growth problem and I'm a bit unsure how to set this up to solve for a solution. Any advice?

Use partial fractions to write 1/(P(P_max-P)) as 1/P_max{1/(P_max-P)+1/P}. Then you can integrate the LHS in terms of natural logs.

 

[Ave22]

Could I get some help with absolute values. I'm to assume abs(x-a) < abs(a)/2 in order to prove that abs(x) > abs(a)/2, with a being a real number. I've done as much as I can moving things about, but I don't know what to do with abs(x-a) regarding how to break it down using a general form.

 

[Leezard]

Could I get some help with absolute values. I'm to assume abs(x-a) < abs(a)/2 in order to prove that abs(x) > abs(a)/2, with a being a real number. I've done as much as I can moving things about, but I don't know what to do with abs(x-a) regarding how to break it down using a general form.

You can always try testing all possible cases. That is, what happens if a is positive or negative, or if x is positive or negative. If you can get the inequality to be true for both positive and negative values for a it shouldn't be any different than regular inequalities.

 

[TimesEunuch]

Could I get some help with absolute values. I'm to assume abs(x-a) < abs(a)/2 in order to prove that abs(x) > abs(a)/2, with a being a real number. I've done as much as I can moving things about, but I don't know what to do with abs(x-a) regarding how to break it down using a general form.

The main thing to note is that abs(x-a) < epsilon (where epsilon>0) is equivalent to saying that -epsilon < x-a < epsilon. Apply this with epsilon = abs(a)/2 (you also need to assume that a is a nonzero number for the inequality to hold).

 

[Ave22]

Thanks for the replies. I'll see what I can do with this.

 

[totowhoa]

I'm in your run-of-the-mill first statistics class, and I have a question. When you are looking at, say, five or six indexes arranged in a class by a price range[15 - 24, 25 - 34, etc], how is it that you determine the median class? Does it have something to do with the relative frequency? Didn't catch what he said in class about how to do it while I was scribbling down some notes. Looking at the info I have and two examples, I'm not seeing how you would determine the median class.

I know it's nothing hard, and I know this isn't really a straight up math problem, but help would be appreciated

 

[Sealda]

How do you solve:


2^x times 3^(x-2) = 4

I want a valuable solution that i can implement on similar kinds of equations.

 

[The_Inquisitor]

How do you solve:


2^x times 3^(x-2) = 4

I want a valuable solution that i can implement on similar kinds of equations.

I'd suggest taking the ln() of both sides and see where that takes you. In fact, I strongly recommend it.

And as a bonus hint

ln(ab) = ln(a) + ln(b)

Is this what you were looking for?

Edit: The answer should be something like...

x = (ln(4)+2ln(3))/(ln(2)+ln(3))

Did this quickly in my head.

 

[Zeppu]

How do you solve:


2^x times 3^(x-2) = 4

I want a valuable solution that i can implement on similar kinds of equations.

Um...

2^x * 3^x * 3^-2 = 4
6^x * 1/9 = 4
6^x = 36 [log both sides. Remember that ln (x^y) =y ln(x)]
x ln(6) = ln(36)
x = 2

Just so you can use this for further questions here is what I used

x^a * x^b = x^(a+b)
x^z * y^z = (xy)^z
ln(x^y) = y ln(x)

 

[Sealda]

I'd suggest taking the ln() of both sides and see where that takes you. In fact, I strongly recommend it.

And as a bonus hint

ln(ab) = ln(a) + ln(b)

Is this what you were looking for?

Edit: The answer should be something like...

x = (ln(4)+2ln(3))/(ln(2)+ln(3))

Did this quickly in my head.

Yes, that works. But its listed under Exponential functions and it gives no clue in the theory how to solve these kinda equations. Its first in next chapter with its theory and exercises they even involve Log and Ln in functions.

Its written in the book as

2^x times 3^x-2= 4

It feels like there should be an "easier" way.

 

[Sealda]

Um...

2^x * 3^x * 3^-2 = 4
6^x * 1/9 = 4
6^x = 36 [log both sides. Remember that ln (x^y) =y ln(x)]
x ln(6) = ln(36)
x = 2

Just so you can use this for further questions here is what I used

x^a * x^b = x^(a+b)
x^z * y^z = (xy)^z
ln(x^y) = y ln(x)

I tried that method before. I thought 2^x times 3^x equaled= 6^2x because the answer i got then was x=1. So i guess that is where i fucked up.

Thank you.

 

[Zeppu]

I tried that method before. I thought 2^x times 3^x equaled= 6^2x because the answer i got then was x=1. So i guess that is where i fucked up.

Thank you.

My pleasure! Feels good to do some maths man! Haven't touched this shit in aaaaaages!

wolframalpha says my answer is correct so don't worry

 

[Mr. Blonde]

The owner has $45
The patrons have $3
The waiter has $2

45 and 3 and 2 is 50. case closed.

 

[The_Inquisitor]

I tried that method before. I thought 2^x times 3^x equaled= 6^2x because the answer i got then was x=1. So i guess that is where i fucked up.

Thank you.

For exponents where its some constant C^X, where x is the variable you want to solve you have to use ln's to unlock the x out of the exponent. What I did above is using pure ln's, but I like how it was simplified first too.

 

[Demokrit]

Its written in the book as

2^x times 3^x-2= 4

It feels like there should be an "easier" way.

Well, of course, 2^x * 3^x = 4 * 3^2 = 2^2 * 3^2 (or 6^x = 6^2 if you prefer), so in this case, the bases being the same, the solution is "obvious" without using logarithms just by comparing the exponents.

 

[Kenka]

Could anybody explain to me in a simple way why the dual spaces are such an important notion in mathematics ? I can't help but scratch my head each time I look on wikipedia. But, I don't really know why, I want to understand.

 

[jepense]

Could anybody explain to me in a simple way why the dual spaces are such an important notion in mathematics ? I can't help but scratch my head each time I look on wikipedia. But, I don't really know why, I want to understand.

It's been a while since I've done functional analysis, but duality in general tends to be important because it is often a tool for connecting two seemingly different structures so that things that are simple(r) to understand in the context of one can be expanded to the other. As an example of a real application: functional analysis is central in quantum mechanics, where physical states form a vector space and linear functionals the dual space. If you know the braket notation, the "bra" form a dual space.

 

[Kenka]

It's been a while since I've done functional analysis, but duality in general tends to be important because it is often a tool for connecting two seemingly different structures so that things that are simple(r) to understand in the context of one can be expanded to the other. As an example of a real application: functional analysis is central in quantum mechanics, where physical states form a vector space and linear functionals the dual space. If you know the braket notation, the "bra" form a dual space.

I'll dig in this direction. Thank you

 

[Government-man]

I'm having a hard time figuring out how this:

(log is the natural logarithm (ln))

is equal to this: