Preface

This is the note of the MIT Course 18.01.1 Calculus. For more details, refer to MIT 18.x Catalog, and it can also be found at MIT Open Learning Library.

Unit 3: Approximations

Linear Approximations of Basic Functions Near 0

$$ \begin{aligned} &(1+x)^r&\approx&\quad 1+r\cdot x\\ &\sin(x)&\approx&\quad \sin(0)+cos(0)\cdot x&=&x\\ &\cos(x)&\approx&\quad\cos(0)-\sin(0)\cdot x&=&1\\ &e^x &\approx&\quad e^0+e^0\cdot x&=&1+x\\ &\ln(1+x)&\approx&\quad\ln(1+0)+\frac 1{1+0}x&=&x \end{aligned} $$

Best Fit Quadratic

The best fit quadratic or best fit parabola to a function $f(x)$ at the point $x=0$ is the quadratic function $q(x)$ whose value agree with the value of $f$ at $x=0$, and those first and second derivatives agree with the first and second derivatives of $f$ at $x=0$.

i.e. $$ \begin{aligned} f(0)\quad &=&q(0)\\ f(0)^\prime\quad &=&q^\prime(0)\\ f(0)^{\prime\prime}\quad &=&q^{\prime\prime}(0) \end{aligned} $$

Quadratic Approximation

The quadratic approximation near $x=a$ is the best fit parabola to $f$ at the point $x=a$.

The formula for the quadratic approximation of a function $f$ near a point $x=a$ is $$ f(x)\approx f(b)+f^\prime (a)\cdot (x-a)+\frac{f^{\prime\prime}(a)}{2}\cdot(x-a)^2 $$ When $a=0$, this quadratic approximation becomes $$ f(x)\approx f(0)+f^\prime(0)\cdot x+\frac{f^{\prime\prime}(0)}2 \cdot x^2 $$

Big-O Notation

A function $f(x)$ is on the order $x^n$ near $x=0$, which is denoted using big “O” notation as $f(x)=O(x^n)$ near $x=0$, if $\vert f(x)\vert \le k\cdot x^n$.

Newton’s Method

Given a function $f(x)$, find $x$ such that $f(x)=0$.

  1. Make a good guess of $x_0$.

  2. Call $x_1$ the $x-$intercept of the tangent line through $(x_0, f(x_0))$. It has the formula $$ x_1=x_0-\frac{f(x_0)}{f^\prime(x_0)} $$

  3. Repeat. The general formula is $$ x_{n+1}=x_n-\frac{f(x_n)}{f^\prime(x_n)} $$ for the $n=0,1,2,\cdots$ .