M6C Data Science I - math review
Magda Gregorova
26/2/2019
2 Basic classes of functions
These notes are reusing text and imagery from: Calculus Volume 1 from OpenStax, Print ISBN 193816802X, Digital ISBN 1947172131, https://www.openstax.org/details/calculus-volume-1 and Precalculus from OpenStax, Print ISBN 1938168348, Digital ISBN 1947172069, https://www.openstax.org/details/precalculus
2.1 Linear
Linear functions have the slope-intercept form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. Its graph is a line.
The slope \(m\) of a line is the change in \(y\) for each unit change in \(x\). \[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \enspace .\] The y-intercpet \(b\) is the function evaluation at \(f(0)\).
The standard form of a line is given by equation \(ax + by = c\), where \(a, b, c\) are constants. This allows for a vertical line (which is not a function).
2.2 Polynomials
Polynomial function is any function that can be written as \[f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \enspace .\] The value \(n\) is called the degree of the polynomial and \(a_n\) the leading coefficient.
Linear function \(f(x) = mx + b\) for \(m \neq 0\) is a polynomial of degree 1 and with \(m = 0\) polynomial of degree 0.
A quadratic function is a polynomial of degree 2 \(f(x) = ax^2 + bx + c\).
A cubic function is a polynomial of degree 3.
- graphs of quadratic functions, (b) graphs of cubic functions
Power functions are of the form \(f(x) = ax^b\), where \(a, b\) are any real numbers.
2.3 Absolute value
Absolute value function is the distance from zero of a number on a real line \[f(x) = |x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases} \]
2.4 Trigonometrics
In math we often measure angles in radians rathe then degrees. Radian is the angle corresponding to the arc of length 1 on a unit circle.
The main trigonometric functions are the \(x\) and \(y\) values of the point on a unit circle. \[\sin t = y \qquad \cos t = x\]
As we travel along the unit circle, the trigonometric functions repeat since \(\theta = \theta + 2\pi\). The trigonometric functions are periodic with a period of \(2\pi\).
2.5 Inverse
Can a function operate in reverse?
Definition: An inverse function \(f^{-1}\) (if it exists) reverses the operation of function \(f\) so that \(f^{-1}(y) = x\) if \(f(x) = y\). In other words, for a function \(f\) and its inverse \(f^{-1}\) we have \[f^{-1}( f(x) ) = x \qquad \text{and} \qquad f(f^{-1}(y)) = y \enspace .\] Note: Generally \(f^{-1}(x) \neq 1 / f(x)\).
There is a relationship between the domain and the range of the function and its inverse.
Careful: A function maps each input to exactly one output. For an inverse function to exist, the original function has to be one-to-one \(f(x_1) \neq f(x_2)\) if \(x_1 \neq x_2\).
2.6 Exponential and logarithmic
2.6.1 Exponential function
The exponential function can be written as \(f(x) = a b^x\), where \(a\) is any non-zero real number and \(b\) is positive real number other then 1.
Careful: Function \(f(x) = ax^b\) is a power function, not exponential.
- Exponential growth is an increase based on a constant multiplicative rate of change over time (percent increase); exponential function grows if \(b > 1\).
- Exponential decay is an decrease based on a constant multiplicative rate of change over time (percent increase); exponential function decays if \(0< b < 1\).
- Linear growth is an increase based on a constant amount over time
| \(x\) | \(f(x) = 2^x\) | \(f(x) = 2x\) | \(f(x) = x^2\) |
|---|---|---|---|
| 0 | 1 | 0 | 0 |
| 1 | 2 | 2 | 1 |
| 2 | 4 | 4 | 4 |
| 3 | 8 | 6 | 9 |
| 4 | 16 | 8 | 16 |
| 5 | 32 | 10 | 25 |
| 6 | 64 | 12 | 36 |
2.6.1.1 Laws of exponents
For any constants \(a, b > 0\) and and all \(x, y\) \[\begin{eqnarray} b^x b^y = b^{x+y} \quad & \quad \frac{b^x}{b^y} = b^{x-y} \quad & \quad (b^x)^y = b^{xy}\\ b^{-x} = \frac{1}{b^x} \quad & \quad (ab)^x = a^x b^x \quad & \quad (\frac{b}{a})^x = \frac{b^x}{a^x} \\ b^{1/x} = \sqrt[x]{b} \quad & \quad \sqrt[x]{ab} = \sqrt[x]{a} \sqrt[x]{b} \quad & \quad \sqrt[x]{\frac{b}{a}} = \frac{\sqrt[x]{b}}{\sqrt[x]{a}} \\ b^{x/y} = \sqrt[y]{b^x} = (\sqrt[y]{b})^x \end{eqnarray}\]
2.6.1.2 Euler’s number \(e\)
\[e = \lim_{n \to \infty} (1 + \frac{1}{n})^n \approx 2.718282\dots\] Function \(f(x) = e^x\) is called the natural exponential function (or often simply exponential) and arrises frequently in practice.
Interestingly, the tangent line of a natural exponential function at point \(x = 0\) has slope 1 (important later).
2.6.2 Logarithmic function
Logarithmic function is the inverse of an exponential function. \[\log_b (x) = y \text{ if and only if } b^y = x\]
We read \(f(x) = \log_b(x)\) as “logarithm (or log) with base \(b\) of \(x\)”.
Because exponential and logarithm and inverses we have \[\log_b (b^x) = x \quad \text{ and } \quad b^{\log_b(x)} = x\]
2.6.2.1 Natural logarithm
The most commonly used logairthmic function is \(\log_e\) with the natural base \(e\). Often it is indicated simply as \(\log\) or \(\ln\).
It is the inverse of the natural exponential function so that \[\ln (e^x) = x \quad \text{ and } \quad e^{\ln(x)} = x\]
2.6.2.2 Important properties
For \(a, b, c > 0\), \(b \neq 1\) and \(r \in \mathbb{R}\) \[\begin{eqnarray} \log_b(ac) & = & \log_b(a) + \log_b(c) \\ \log_b(a/c) & = & \log_b(a) - \log_b(c) \\ \log_b(a^r) & = & r\log_b(a) \end{eqnarray}\]