04-Chapter_4.Rmd

# (PART) Review of math and statistics {-}

# Basic mathematical tools {#chapter4}

## The summation operator {.unnumbered}

## Exponents {.unnumbered}

## Equations {.unnumbered}

## Simultaneous equations {.unnumbered}

R can perform standard matrix algebra operations. We can use matrix algebra functions in R to solve our problem from class.

$$
7x + 5y - 3z = 16 \\
3x -5y +2z	= -8 \\
5x +3y -7z = 0
$$

First, we rewrite the system using matrix and vector notation:

$$
\mathbf{A} = \left[\begin{array}
{rrr}
7 & 5 & -3 \\
3 & -5 & 2 \\
5 & 3 & -7 \\
\end{array}\right]
\mathbf{b} = \left[\begin{array}
{rrr}
x \\
y \\
z \\
\end{array}\right]
\mathbf{r} = \left[\begin{array}
{rrr}
16 \\
-8 \\
0 \\
\end{array}\right]
$$

In order to obtain the result vector *b*, we have to rearrange the model performing some simple matrix algebra operations.

$$
\mathbf{A}^{-1}\mathbf{A}\mathbf{b} = \mathbf{A}^{-1}\mathbf{r} \\
\text{remember that} \space \mathbf{A}^{-1}\mathbf{A} = \mathbf{I} \\
\mathbf{b} = \mathbf{A}^{-1}\mathbf{r}
$$ 

We are now ready to solve our system of equations using R:

```{r}
data <- c(7, 5, -3, 3, -5, 2, 5, 3, -7)

A <- matrix(data, nrow = 3, ncol = 3, byrow = TRUE)

r <- c(16, -8, 0)

b <- solve(A) %*% r

b
```

## Logarithms {.unnumbered}

## Derivatives {.unnumbered}

## Optimization {.unnumbered}

## Derivative rules {.unnumbered}

**The rules for sums and differences**

Given $ f(x) = g(x) \pm h(x) $, where $g(x)$ and $h(x)$ are both differentiable functions, the derivative of a sum or difference of two functions is given by:

\begin{equation}
f'(x) = g'(x) \pm h'(x)
\end{equation}

**The product rule**

Given $f(x) = g(x) \cdot h(x)$, where $g(x)$ and $h(x)$ are both differentiable functions, the derivative is given by:  

\begin{equation}
f'(x) =  h(x) \cdot g'(x) + h'(x) \cdot g(x)
\end{equation}

**The quotient rule** 

Given $f(x) = \frac{g(x)}{h(x)}$, where $g(x)$ and $h(x)$ are both differentiable functions and $h(x)\neq0$, the derivative is given by:  

\begin{equation}
f'(x) =  \dfrac{h(x) \cdot g'(x) - h'(x) \cdot g(x)}{[h(x)]^2}
\end{equation}

**The generalized power function rule**

Given $f(x) = [g(x)]^n$, where $g(x)$ is a differentiable functions and n is any real number, the derivative is given by: 

\begin{equation}
f'(x) =  n[g(x)]^{n-1} \cdot g'(x)
\end{equation}

**The chain rule**

Given $f(x) = h(g(x))$, where $f$ is a function of a function where $h$ is in turn function of \textit{g}, the derivative is given by:

\begin{equation}
f'(x) = h'(g(x)) \cdot g'(x)
\end{equation}

**Additional rules**

Given $f(x) = e^x$, the derivative is given by:

\begin{equation}
f'(x) = e^x
\end{equation}

Given $f(x) = ln(x)$, the derivative is given by:

\begin{equation}
f'(x) = \frac{1}{x}
\end{equation}

Given $f(x) = sin(x)$, the derivative is given by:

\begin{equation}
f'(x) = cos(x)
\end{equation}

Given $f(x) = cos(x)$, the derivative is given by:

\begin{equation}
f'(x) = -sin(x)
\end{equation}