Math221 Fall06, Professor Ram

Definitions

The positive integers are 1, 2, 3, 4, 5, 6, ...

The nonnegative integers are 0, 1, 2, 3, 4, 5, 6, ...

The integers are −3, −2, −1, 0, 1, 2, 3, ...

The rational numbers are

$\frac{a}{b}$, where $a$ is an integer and $b$ is an integer which is not equal to 0

The real numbers are all possible decimal expansions.

The complex numbers are $a+bi$ where $a$ and $b$ are real numbers and $i$ is a number such that$i^2=-1$.

If $n$ is a positive integer then $n$-factorial is

If $n$ is a positive integer then

$a^n=\underset{\mathrm{nfactors}}{a.a....a}$

This satisfies $a^{x+y}=a^x{a}^y$ and $a^1=a$. These two relations are enough to derive that

$a^x=e^{x\ln a}$, where $e^{\ln a}=a$.

A function $f$ is a creature that eats a number, chews on it, and spits out a new number.

What a function spits out depends only on what goes in.

The inverse of a function, $f^{-1}$, is $f$ backwards.

The inverse of a function is not always a function because there may be some uncertainty about what the inverse function will spit out.

The exponential function is

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+···$

The sine function is

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-···$

The cosine function is

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-$

The hyperbolic sine function is

$\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\frac{x^9}{9!}-···$

The hyperbolic cosine function is

$\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}+$

The tangent, cotangent, secant and cosecant functions are

$\tan x=\frac{\sin x}{\cos x}$, $\cot x=\frac{1}{\tan x}$, $\sec x=\frac{1}{\cos x}$, $\csc x=\frac{1}{\sin x}$.

The hyperbolic tangent, cotangent, secant and cosecant functions are

$\tanh x=\frac{\sinh x}{\cosh x}$, $\coth x=\frac{1}{\tanh x}$, $\mathrm{sech}x=\frac{1}{\cosh x}$, $\mathrm{csch}x=\frac{1}{\sinh x}$.

Then

$\sqrt{x}$,$\ln x$, ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$, ${\cot }^{-1}x$, ${\sec }^{-1}x$,${\csc }^{-1}x$, ${\sinh }^{-1}x$, ${\cosh }^{-1}x$, ${\tanh }^{-1}x$, ${\coth }^{-1}x$, ${\mathrm{sech}}^{-1}x$, ${\mathrm{csch}}^{-1}x$

are the inverse functions to $x^2$, $e^x$, $\sin x$, $\cos x$, $\tan x$, $\cot x$, $\sec x$, $\csc x$, $\sinh x$, $\cosh x$, $\tanh x$, $\coth x$, $\mathrm{sech}x$, and $\mathrm{csch}x$, respectively.

A derivative is a creature who eats a function, chews on it, and spits out a new function

A integral is the derivative backwards

or