Topics within Calculus
Before you go off to derivatives (oh, how much fun) there's a few more things you should know.
If you skip this, and I know it's tempting, you'll be confused later. I'll reference this page later, so you better read it now instead of coming back to it when you've already started derivatives.
That said, these are really just some really nice and simple answers to common problems, just like when you learned in trigonometry that $\sin{30}=\frac{1}{2}$. (You did, right?)
IS#
$$\ln{(x+1)}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots\pm\frac{x^{\infty}}{\infty}$$
$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots+\frac{x^{\infty}}{\infty!}$$
LI#
$$\lim_{x\to 0}\frac{a^{x}-1}{x}=\ln{a}$$
$$\lim_{x\to 0}\frac{e^x-1}{x}=1$$
$$\lim_{x\to 0}\frac{\sin{x}}{x}=1$$
$$\lim_{x\to 0}\frac{\cos{x}-1}{x}=0$$
$$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e$$
Can you use LI1
to prove LI2
? How about LI3
to prove LI4
?
(Hint: Yes you can, now do it!)