How to find the local maxima and minima
How many local maxima and local minima can you find in the cover image? Is there a systematic way to find all maxima and minima of a function? In this HowDo we show how to find the extrema of a function. We shall get information from the derivative $f'(x)$ is to find the local maxima and minima. We shall assume the function $f(x)$ is generally differentiable but may possess a number of singular points. Various examples come near the end of this HowDo.
How to sketch the graph of a function
Here we show how to sketch the graph of a function $f(x)$, using information obtained from the function itself and its derivatives. We may classify all the features according to its source, into zeroth, first, second order considerations. It is impossible to get all the details correct, but it will help us get a sense of what the curve looks like. In this HowDo, we will use $f(x) = x^{3}-3x-1$ as an example.
How to determine whether a function is even or odd (or neither)
Intuitively, a function is even if its graph is symmetric with respect to the $y$-axis (Imagine putting a mirror on the $y$-axis, the left side is the reflection of the right side). See the cover image for some examples of even functions. On the other hand, a function is odd if its graph is symmetric with respect to the origin. In terms of symbols, the condition that a function $f$ is even can be expressed as $f(-x) = f(x).$A function $f$ is odd if $f(-x) = -f(x).$
How to find the Taylor Polynomial of a function
The $n$-th degree Taylor Polynomial $T_n$ of a function $f(x)$ at a given point $x = a$ is defined as the polynomial that is closest to the function in an infinitesimally small neighbourhood of $x = a$. There is a standard formula $T_n(x) = f(a) + f'(a) (x-a) + \frac{f''(a)}{2!} (x-a)^2 +\cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n.$ In this HowDo, I will explain how to find the Taylor Polynomial in question using this formula. We will calculate the Taylor Polynomial of $f(x) = \sin{x}$ at $x = \frac{\pi}{3}$ to $3$rd degree as example.
How to find the limit of a Rational function at infinity
A rational function is defined as the ratio of two polynomials. In general, we may write $R(x) = \frac{P(x)}{Q(x)}, $ where $P$ and $Q$ are polynomials. In this HowDo, we show how to find the limit of a rational function at positive or negative infinity. That is, we are interested in finding $\lim_{x\to\infty}R(x) = \lim_{x\to\infty}\frac{P(x)}{Q(x)}$ and $\lim_{x\to-\infty} R(x) = \lim_{x\to-\infty}\frac{P(x)}{Q(x)}.$We shall use $R(x) = \frac{x^3-x^2+2x+4}{ x -x^4 +1}$ as an example. At the end we will also provide examples to illustrate all cases mentioned in the HowDo.
How to find the tangent line (linear approximation) of a function
In this HowDo, you will learn how to find the equation of the tangent line of a function $f(x)$ at a point $x = a$. Sometimes, we say that the tangent line is the linear approximation of the function $f(x)$ near $x=a$. As an example, we shall find the derivative of $f(x) = x^2 - 2$ at the point $x = 2$. The equation of the tangent will be found to be $y = 4x-6$. So one may say that $y = 4x-6$ is the linear approximation of $f(x) = x^2 - 2$ near $x = 2$.
How to use Newton's method to find the root of a function
Here we demonstrate how to use Newton's method to find increasingly accurate guesses of the root of a function. We use $f(x) = x^2 - 2$ as an example. We know that the root of this function is $\sqrt{2}$, which is approximately $1.414$. We can make an initial guess of $x_0 = 1$, and use Newton's method to improve upon the guess.