math117: add up to pfd

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eggy 2022-09-23 11:01:07 -04:00
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@ -43,7 +43,7 @@ If a function is not invertible, restricting the domain may allow a **partial in
By restricting the domain to $[0,\inf]$, the **multivalued inverse function** $y=\pm\sqrt{x}$ is reduced to just the partial inverse $y=\sqrt{x}$.
## Symmetry
78u7u887878
An **even function** satisfies the property that $f(x)=f(-x)$, indicating that it is unchanged by a reflection across the y-axis.
An **odd function** satisfies the property that $-f(x)=f(-x)$, indicating that it is unchanged by a 180° rotation about the origin.
@ -70,3 +70,133 @@ $$
\sinh x = \frac{1}{2}(e^x - e^{-x})
$$
## Piecewise functions
A piecewise function is one that changes formulae at certain intervals. To solve piecewise functions, each of one's intervals should be considered.
### Absolute value function
$$
\begin{align*}
|x| = \begin{cases}
x &\text{ if } x\geq 0 \\
-x &\text{ if } x < 0
\end{cases}
\end{align*}
$$
### Signum function
The signum function returns the sign of its argument.
$$
\begin{align*}
\text{sgn}(x)=\begin{cases}
-1 &\text{ if } x < 0 \\
0 &\text{ if } x = 0 \\
1 &\text{ if } x > 0
\end{cases}
\end{align*}
$$
### Ramp function
The ramp function makes a ramp through the origin that suddenly flatlines at 0. Where $c$ is a constant:
$$
\begin{align*}
r(t)=\begin{cases}
0 &\text{ if } x \leq 0 \\
ct &\text{ if } x > 0
\end{cases}
\end{align*}
$$
<img src="https://upload.wikimedia.org/wikipedia/commons/c/c9/Ramp_function.svg" width=700>(Source: Wikimedia Commons, public domain)</img>
### Floor and ceiling functions
The floor function rounds down.
$$\lfloor x\rfloor$$
The ceiling function rounds up.
$$\lceil x \rceil$$
### Fractional part function
In a nutshell, the fractional part function:
- returns the part **after the decimal point** if the number is positive
- returns 1 - **the part after the decimal point** if the number is negative
$$\text{FRACPT}(x) = x-\lfloor x\rfloor$$
Because this function is periodic, it can be used to limit angles to the $[0, 2\pi)$ range with:
$$f(\theta) = 2\pi\cdot\text{FRACPT}\biggr(\frac{\theta}{2\pi}\biggr)$$
### Heaviside function
The Heaviside function effectively returns a boolean whether the number is greater than 0.
$$
\begin{align*}
H(x) = \begin{cases}
0 &\text{ if } t < 0 \\
1 &\text{ if } t \geq 0
\end{cases}
\end{align*}
$$
This can be used to construct other piecewise functions by enabling them with $H(x-a)$ as a factor, where $a$ is the interval.
In a nutshell:
- $1-H(t-a)$ lets you "turn a function off" at at $t=a$
- $H(t-a)$ lets you "turn a function on at $t=a$
- $H(t-a) - H(t-b)$ leaves a function on in the interval $(a, b)$
!!! example
TODO: example for converting piecewise to heaviside via collecting heavisides
and vice versa
## Periodicity
The function $f(t)$ is periodic only if there is a repeating pattern, i.e. such that for every $x$, there is an $f(x) = f(x + nT)$, where $T$ is the period and $n$ is any integer.
### Circular motion
Please see [SL Physics 1#6.1 - Circular motion](/g11/sph3u7/#61-circular-motion) and its subcategory "Angular thingies" for more information.
## Partial function decomposition (PFD)
In order to PFD:
1. Factor the denominator into irreducibly quadratic or linear terms.
2. For each factor, create a term. Where capital letters below are constants:
- A linear factor $Bx+C$ has a term $\frac{A}{Bx+C}$.
- A quadratic factor $Dx^2+Ex+G$ has a term $\frac{H}{Dx^2+Ex+G}$.
3. Set the two equal to each other such that the denominators can be factored out.
4. Create systems of equations to solve for each constant.
!!! example
To decompose $\frac{x}{(x+1)(x^2+x+1)}$:
$$
\begin{align*}
\frac{x}{(x+1)(x^2+x+1)} &= \frac{A}{x+1} + \frac{Bx+C}{x^2+x+1} \\
&= \frac{A(x^2+x+1) + (Bx+C)(x+1)}{(x+1)(x^2+x+1)} \\
x &= A(x^2+x+1) + (Bx+C)(x+1) \\
0x^2 + x + 0 &= (Ax^2 + Bx^2) + (Ax + Bx + Cx) + (A + C) \\
\\
&\begin{cases}
0 = A + B \\
1 = A + B + C \\
0 = A + C
\end{cases}
\\
A &= -1 \\
B &= 1 \\
C &= 1 \\
\\
∴ \frac{x}{(x+1)(x^2+x+1)} &= -\frac{1}{x+1} + \frac{x + 1}{x^2 + x + 1}
\end{align*}
$$