Calculus Watch

roshanhero
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I am starting this new subject calculus,but there are already lots of problem that I have been facing so,please enlighten me.
1.I generally understood the definition of LIMITS ,but I couldnot understand the graphical representation or meaning of limts.
2.Also,I couldnot understand the meaning of bounded functions especially, graphically.
3.I am completely messed up with the monotonic functions with all those weird notations,so please explain me about this more.
Thanks.............
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yusufu
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How advanced are you in your study?
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roshanhero
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Undergraduate level
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Zhen Lin
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Wait, I thought you were just taking A level Further Mathematics just a few weeks ago.
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pyrolol
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Surely part of understanding the definition of a limit involves a graphical understanding?

Ok - what does the definition say, in slightly graphical terms? That, approaching a point, you can find a region so that the curve is always within <something> of the value of the limit. That is, there is a little region around a point (well, depends if you're approaching from one specific side etc.) where the value is as close as you want to the limit. (Obviously what happens at the point itself is irrelevant).

A bound is just a number the funtion never exceeds (or never is less than) (That is, a value . Imagine a line, let's say y = 10, and then the equation of y = arctanx. It's clear that arctan never exceeds 10, so 10 is an upper bound on it (not a very good one, but that's not the point.) You can do a similar thing for a lower bound (y=-10). (NB: all left hand sides of / correspond etc.) A least/greatest upper/lower bound (supremum/infimum), is a upper/lower bound, such that any number less/greater than it is not a upper/lower bound - that is, you can't 'improve' on it. +-pi/2 work nicely with the aforementioned example. Note that arctanx never actually equals them, but that they're its limits as x->+-infinity respectively. I think I'm right in saying that for a continuous function, the infimum/supremum are either acheived at a turning point, or at a limit (example of latter: inf and sup of arctan are at it's limits, former: consider (x^2-1)e^(-x^2).)

I've had a quick look and wikipedia have some nice diagrams for monotonicity.
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roshanhero
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(Original post by Zhen Lin)
Wait, I thought you were just taking A level Further Mathematics just a few weeks ago.
No,I am doing B.Sc in physics in my country.
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roshanhero
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Thanks.
I know about two methods of solving integration.They are
substitution and integration by parts.Can you tell me about other methods as well like Integration by decomposition into a sum and integration by successive reduction.
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pyrolol
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The latter is usually based on integration by parts. Let's say you wanted to integrate I_n(x) = \int{\sin{x}^{n}}dx What you would do, is split it into sin^2 * sin^(n-2), write the former as 1 - cos^2, use the distributive property to split it into two integrals, one of which is I_(n-2)(x). The second part can be integrated by parts, taking u as cosx, and dv/dx as cosxsin^(n-2)x. You'll see that you can rearrange it into an equation relating I_n to I_(n-2). This makes it easy for you to find something like I_7 between 0 and pi/2 (try it!).

The other type mostly involve using parts directly.

Here's a rational function f(x) = \frac{5x^4+8x^3+8x^2-6x+1}{(x^2+1)(x-2)(x^2+2x+3)}
You know how to split this into partial fractions? (There's one other form I missed, where you have a (x+a)^n for n >= 2; what happens here is you either write it as a quadratic (/cubic etc) with a general polynomial degree one lower on top, or you split it into
Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\sum_{i=1}^n{\frac{c_i}{(x+a)^i}
. Also if the fraction is improper (degree of numerator >= degree of denominator), you need to use algebraic division to write it as a polynomial + a proper fraction.

Now you integrate each term. I've tried to put a few different techniques in the above example, you should have two parts become logarithms, and one require a linear then trig substitution. Obviously there are more, I assume you have example to work through?
f(x) = \frac{Ax+B}{x^2+1} + \frac{C}{x-2} + \frac{Dx+E}{x^2+2x+3}
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roshanhero
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(Original post by pyrolol)
The latter is usually based on integration by parts. Let's say you wanted to integrate I_n(x) = \int{\sin{x}^{n}}dx What you would do, is split it into sin^2 * sin^(n-2), write the former as 1 - cos^2, use the distributive property to split it into two integrals, one of which is I_(n-2)(x). The second part can be integrated by parts, taking u as cosx, and dv/dx as cosxsin^(n-2)x. You'll see that you can rearrange it into an equation relating I_n to I_(n-2). This makes it easy for you to find something like I_7 between 0 and pi/2 (try it!).

The other type mostly involve using parts directly.

Here's a rational function f(x) = \frac{5x^4+8x^3+8x^2-6x+1}{(x^2+1)(x-2)(x^2+2x+3)}
You know how to split this into partial fractions? (There's one other form I missed, where you have a (x+a)^n for n >= 2; what happens here is you either write it as a quadratic (/cubic etc) with a general polynomial degree one lower on top, or you split it into
Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\sum_{i=1}^n{\frac{c_i}{(x+a)^i}
. Also if the fraction is improper (degree of numerator >= degree of denominator), you need to use algebraic division to write it as a polynomial + a proper fraction.

Now you integrate each term. I've tried to put a few different techniques in the above example, you should have two parts become logarithms, and one require a linear then trig substitution. Obviously there are more, I assume you have example to work through?
f(x) = \frac{Ax+B}{x^2+1} + \frac{C}{x-2} + \frac{Dx+E}{x^2+2x+3}
Thanks to pyrolol a lot,but have you got any links of the sites that provide brilliant tutorials and questons on these topics of integration.
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pyrolol
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http://www.math.ucdavis.edu/~kouba/C...rtialFrac.html

I didn't find anything particularly good on reduction formulae.
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