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MathsIsFun

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Registered: 2005-01-21

Posts: 7,713

Calculus (General) Formulas

Calculus (General) Formulas

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Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: Calculus (General) Formulas

Last edited by Ricky (2006-04-04 04:25:21)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,390

Re: Calculus (General) Formulas

If x=f(t), then dy/dx = (dy/dt)/(dx/dt)

dy/dx*dx/dy=1 or dx/dy = 1/(dy/dx)

d(logx)/dx = 1/x

d(Sinx)/dx = Cosx

d(Cosx)/dx = -Sinx

d(tanx)/dx = Sec²x

d(cotx)/dx = -Cosec²x

d(secx)/dx = secxtanx

d(cosecx)/dx = -CosecxCotx

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: Calculus (General) Formulas

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Calculus (General) Formulas

In the last Ricky post there is a little mistake:
1. lim a f(x)=a m, and
4. lim f(x)/g(x)= m/n , n != 0.
There won't be any eroors if, instead of:
"Let lim f(x)=m and lim g(x)=n",
we write:
"Let lim f(x)=n and lim g(x) = m".

//After the correction remove this

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Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,390

Re: Calculus (General) Formulas

Fundamental limits

, a>0

,a>0

, a>0

,
a>0, n∈N

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Calculus (General) Formulas

Hold it, the limit of tan 1/x as x approaches zero is pi/2? That doesn't seem right.

Shouldn't it be, limit of arctan 1/x as x approaches zero equals pi/2?

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A logarithm is just a misspelled algorithm.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Calculus (General) Formulas

and another fomula say it's -x/2 ???

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Calculus (General) Formulas

limit as x approaches 0 of tan(1/x) is undefined.

Same goes with arctan.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,390

Re: Calculus (General) Formulas

[Thank you, Mikau, George and Ricky. I have incorporated the correction.]

Theorems on limits

where p and q are integers.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,390

Re: Calculus (General) Formulas

Some Important Expansions:-

for -1<x<1

for every positive value of x<1.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Calculus (General) Formulas

limit as x approaches 0 of tan(1/x) is undefined.

Same goes with arctan.

I don't think thats correct. Yes the limit of tan (1/x) as x approaches zero is undefined, but the limit of arctan (1/x) as x approaches zero IS pi/2. Think about it, if  θ is an angle in a right triangle, and the side opposite  θ is 1 and the side adjacent to θ is x, as x approaches zero,  θ aproaches 90 from the left.

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A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Calculus (General) Formulas

This isn't rigourous, but more of a way to think about it.  limit 1/x as x goes to 0 from the positive side is positive infinity.  So what we really want to find is the limit of arctan(x) as x goes to infinity, which is pi/2.

But what about when we approach 1/x from the negative side?  Then it's negative infinity, and thus, we want to find the limit of arctan(x) as x approaches negative infinity.  That's -pi / 2. 

Since the limit from the left is not the limit from the right, the limit does not exist.

For a clear way to see it, just use a graphing calculator.  Or just do arctan(1/.000000001) and arctan(1/-000000001).

Last edited by Ricky (2006-04-10 07:25:59)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Calculus (General) Formulas

Oh, right. Sorry, I forgot no one specified which side it approaches zero from so the left hand limit has to equal the right hand limit, for a limit to exist.

My bad.

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A logarithm is just a misspelled algorithm.

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Calculus (General) Formulas

Maclaurin/Taylor Series

Exponentials and Logarithms

Trigonometric Functions

See bottom of post for definition of B[sub]n[/sub].

See bottom of post for definition of B[sub]n[/sub].

See bottom of post for definition of E[sub]n[/sub].

See bottom of post for definition of B[sub]n[/sub].

Hyperbolic Functions

See bottom of post for definition of B[sub]n[/sub].

See bottom of post for definition of B[sub]n[/sub].

See bottom of post for definition of E[sub]n[/sub].

See bottom of post for definition of B[sub]n[/sub].

B[sub]n[/sub]: The Bernoulli Numbers

The Bernoulli numbers B[sub]n[/sub] are a sequence of special rational numbers. Their derivation is outside the scope of this thread, but an explanation may be offered elsewhere upon sufficient demand. The first few Bernoulli numbers are(note that for odd n other than 1, B[sub]n[/sub] = 0):

E[sub]n[/sub]: The Euler Numbers

The Euler numbers E[sub]n[/sub] are a sequence of special numbers. Their derivation is outside the scope of this thread, but an explanation may be offered elsewhere upon sufficient demand. The first few Euler numbers are(note that for all odd n, E[sub]n[/sub] = 0):

Last edited by Zhylliolom (2006-08-06 22:51:43)

glenn101

Member

Registered: 2008-04-02

Posts: 108

Re: Calculus (General) Formulas

Finding Derivatives By First Principles

f'(x)=lim    f(x+h)-f(x)
        h->0 -------------
                      h

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