Nick and Hippo and Tweetjazz, I would memorize the Taylor series and the general term for sin(x), cos(x), ln(x+1), and e^(x). You may be able to derive them but it is much much MUCH easier to memorize the formula. It also saves you a lot of time.
The way I remember sin(x) and cos(x) is somewhat interesting and I'll share but bare with me.
on the unit circle cos=x right? well, to make an X you need two lines. Two is even so the powers of the maclaurin series for cos(x) are even. and let's pretend that 0 is also even.
so, cos(x)=1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ....
b/c x^0=1
on the unit circle, sin=Y and to make a capital Y, you need three lines. three is odd so the powers for sin are odd.
sin(x)= x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ....
you always divide by the factorial of the power and the series alternate. I thought it was clever but my calculus teacher didn't get it. Hope it helps someone else!
e^x is always simple. it's just all the powers/ the power factorial. you start at 0 and keep going.
so...
e^x= 1 + x +(x^2)/2! + (x^3)/3! + (x^4)/4! + .....
ln(x + 1) may trip you up because there are no factorials in it. The way I think of it is that ln(x+1) has a +1 attached to the X so it's a tricky ricky type situation. So, ln(x+1) is also a tricky ricky situation because it doesn't have the factorials AND it starts at X because X is sexy AND it alternates. (laugh all you want, but I remember these things.
)
so,
ln(x + 1) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + .......
so enjoy. and i hoped you at least got a laugh (if my awesome mind trick didn't help you.)
PS Is it bad that I haven't started studying yet?
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