Mathematical analysis notes

1. Mean value theorems

Theorem 1.1. ($\color{red}{\textrm{Rolle's Theorem}}$) Let $f$ be a function that satisfies the following conditions:

1. $f$ is ${\color{Cyan}{\textrm{continuous}}}$ on the ${\color{orange}{\textrm{closed}}}$ interval $[a,b]$.
2. $f$ is ${\color{Cyan}{\textrm{differentiable}}}$ on the ${\color{orange}{\textrm{open}}}$ interval $(a,b)$.
3. $f(a) = f(b)$

Then there exists $\zeta\in(a,b)$ such that $f'(\zeta) = 0$.

Theorem 1.2. ($\color{red}{\textrm{The Mean Value Theorem}}$) Let $f$ be a function that satisfies the following conditions:

1. $f$ is ${\color{Cyan}{\textrm{continuous}}}$ on the ${\color{orange}{\textrm{closed}}}$ interval $[a,b]$.
2. $f$ is ${\color{Cyan}{\textrm{differentiable}}}$ on the ${\color{orange}{\textrm{open}}}$ interval $(a,b)$.

Then there exists $\xi\in[a,b]$ such that $$ f'(\xi) = \frac{f(b)-f(a)}{b-a} $$ or, equivalently, $$ f(b)-f(a)=f'(\xi)(b-a) $$

Theorem 1.3. ($\color{red}{\textrm{Cauchy's Mean Value Theorem}}$) Suppose that the functions $f$ and $g$ are ${\color{Cyan}{\textrm{continuous}}}$ on $[a,b]$ and ${\color{Cyan}{\textrm{differentiable}}}$ on $(a,b)$, then there exists $\varsigma\in (a,b)$ such that $$ (f(b)-f(a))g'(\varsigma) = (g(b)-g(a))f'(\varsigma) $$

If, in addition, $g(a)\neq g(b)$ and $g'(\varsigma)\neq 0$, this is equivalent to: $$\frac{f'(\varsigma)}{g'(\varsigma)} = \frac{f(b)-f(a)}{g(b)-g(a)}.$$

Theorem 1.4. ($\color{red}{\textrm{Mean Value Theorem For Integrals}}$) If $f$ is continuous on $[a,b]$, then there ${\color{blue}{\textrm{exists}}}$ a number $\varepsilon$ in $[a,b]$ such that $$ \int_{a}^{b}f(x)dx = f(\varepsilon)(b-a)$$

Definition 1.5. The ${\color{DeepPink}{\textrm{mean of a function}}}$ over the interval $[a,b]$ is $$ f(\xi)=\frac{1}{b-a}\int_{a}^{b}f(x)dx $$

Theorem 1.6. ($\color{red}{\textrm{The Fundamental Theorem Of Calculus}}$) If $f$ is ${\color{Cyan}{\textrm{continuous}}}$ on $[a,b]$, then the function $$ \Phi(x)=\int_{a}^{x}f(t)dt $$ is ${\color{Cyan}{\textrm{differentiable}}}$ on $[a,b]$ and its ${\color{orange}{\textrm{derivative}}}$ is $$ \Phi'(x)= \frac{d}{dx}\int_{a}^{x}f(t)dt = f(x) \  \ (a\leq x\leq b).$$

2. The maximum and minimum of a function

Definition 2.1. Let $f$ be a function, and $\zeta$ a number in its domain. Then the number $f(\zeta)$ is a

1. ${\color{red}{\textrm{local maximum}}}$ value of $f$ if $f(\zeta)\geq f(x)$ where $x$ is in a ${\color{orange}{\textrm{neighborhood}}}$ of $\zeta$.
2. ${\color{red}{\textrm{local minimum}}}$ value of $f$ if $f(\zeta)\leq f(x)$ where $x$ is in a ${\color{orange}{\textrm{neighborhood}}}$ of $\zeta$.

Definition 2.2. Let $f$ be a function, and $\zeta$ a number in its domain $\mathds{D}$. Then the number $f(\zeta)$ is a

1. ${\color{red}{\textrm{absolute or global maximum}}}$ value of $f$ if $f(\zeta)\geq f(x)$ for all $x\in\mathds{D}$.
2. ${\color{red}{\textrm{absolute or global minimum}}}$ value of $f$ if $f(\zeta)\leq f(x)$ for all $x\in\mathds{D}$.

Definition 2.3. The ${\color{magenta}{\textrm{maximum and minimum values}}}$ of $f$ are called ${\color{magenta}{\textrm{extreme values}}}$ of $f$.

Remark 2.4. A ${\color{blue}{\textrm{Stationary point}}}$ of $f$ is a point $\zeta$ where $f'(\zeta) = 0$. A ${\color{blue}{\textrm{critical point}}}$ of $f$ is a point $\delta$ such that $f'(\delta) = 0$ or $f'(\delta)$ does not exist. The value of the function at a critical point is called ${\color{blue}{\textrm{critical value}}}$.

Theorem 2.5. ($\color{magenta}{\textrm{Fermat's Theorem}}$) If $f$ has a ${\color{orange}{\textrm{local maximum}}}$ or a ${\color{orange}{\textrm{local minimum}}}$ at $\xi$ and if $f'(\xi)$ exists, then $f'(\xi) = 0$.

Proposition 2.6. ($\color{magenta}{\textrm{The First Derivative Test}}$) Suppose that $\xi$ is a ${\color{Cyan}{\textrm{critical point}}}$ of a ${\color{blue}{\textrm{continuous}}}$ function $f$.

1. If $f'$ changes from positive to negative at $\xi$, then $f$ has a ${\color{red}{\textrm{local maximum}}}$ at $\xi$.
2. If $f'$ changes from negative to positive at $\xi$, then $f$ has a ${\color{red}{\textrm{local minimum}}}$ at $\xi$.
3. If $f'$ does not change sign at $\xi$ (for example, if $f'$ is positive on both sides of $\xi$ or negative on both sides), then $f$ has ${\color{red}{\textrm{no local maximum or minimum}}}$ at $\xi$.

Proposition 2.7. ($\color{magenta}{\textrm{The Second Derivative Test}}$) Suppose $f''$ is ${\color{blue}{\textrm{continuous}}}$ near $\xi$.

1. If $f'(\xi) = 0$ and $f''(\xi)>0$, then $f$ has a ${\color{Cyan}{\textrm{local minimum}}}$ at $\xi$.
2. If $f'(\xi) = 0$ and $f''(\xi)<0$, then $f$ has a ${\color{Cyan}{\textrm{local maximum}}}$ at $\xi$.

Definition 2.8. An ${\color{magenta}{\textrm{inflection point}}}$, ${\color{magenta}{\textrm{point of inflection}}}$, or ${\color{magenta}{\textrm{inflexion}}}$ of $f$ is a point $\xi$ such that $f''(\xi)=0$ and $f''$ ${\color{Cyan}{\textrm{changes sign}}}$ at $\xi$.

3. Asymptotes

Definition 3.1. If $\lim\limits_{x\to{+\infty}}f(x)=L$ or $\lim\limits_{x\to{-\infty}}f(x)=L$, we say the line $y = L$ is a ${\color{RoyalBlue}{\textbf{horizontal asymptote}}}$ of $f$. If $\lim\limits_{x\to{L}}f(x)=\infty$ (or $\lim\limits_{x\to{L^{+}}}f(x)=\infty$, $\lim\limits_{x\to{L^{-}}}f(x)=\infty$), we say the line $x = L$ is a ${\color{Salmon}{\textbf{vertical asymptote}}}$.

$\require{empheq}\begin{empheq}{align*} &\lim\limits_{x\to{+\infty}}[f(x)-(kx+b)]=0 \Leftrightarrow \lim\limits_{x\to{+\infty}}[f(x)-kx]=b.   \\ &\lim\limits_{x\to{+\infty}}[\frac{f(x)}{x}-k]=\lim\limits_{x\to{+\infty}}\frac{1}{x}[f(x)-kx]=0\cdot b=0 \Rightarrow \lim\limits_{x\to{+\infty}}\frac{f(x)}{x}=k \end{empheq}$

Then the line $y=kx+b$ is an ${\color{olive}{\textrm{oblique asymptote}}}$ or an ${\color{Sepia}{\textrm{slant asymptote}}}$ of $f$.

4. Functions in several variables

Theorem 4.1. $\color{magenta}{\textrm{The Chain Rule (Case 1).}}$ Suppose that ${\color{red}{z = f(x,y)}}$ is a ${\color{Cyan}{\textrm{differentiable function}}}$ of $x$ and $y$, where ${\color{red}{x=g(t)}}$ and ${\color{red}{y=h(t)}}$ are both ${\color{Cyan}{\textrm{differentiable functions}}}$ of $t$. Then $z$ is a ${\color{Cyan}{\textrm{differentiable function}}}$ of $t$ and

$\begin{align*}\fbox{ $\displaystyle \frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$ } \end{align*}$

Definition 4.2. The ${\color{red}{\textbf{total differential}}}$ $dz$ is defined by

$$\displaystyle dz = f_{x}(x,y)dx+f_{y}(x,y)dy=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial x}dy $$

Theorem 4.3. If $F(x,y)$ is defined on a disk $\mathds{D}$ containing $(a,b)$, where $F(a,b) = 0$, $F_{y}(a,b)\neq 0$, and $F_{x}$ and $F_{y}$ are continuous on $\mathds{D}$, then the equation $F(x,y) = 0$ defines a function $y=f(x)$ near $(a,b)$ such that

$\begin{align*}\displaystyle \frac{dy}{dx}=-\frac{\partial F}{\partial x}\bigg/\frac{\partial F}{\partial y}=-\frac{F_{x}}{F_{y}} \end{align*}$

5. Equivalent infinitesimal

$\begin{align*} \sin x&\sim x   \\ \tan x&\sim x  \\ \arctan x&\sim x \\ \arcsin x&\sim x \\ 1-\cos x&\sim \frac{1}{2}x^{2} \\ \ln(1+x)&\sim x \\ e^{x}-1&\sim x \\ \sqrt[n]{(1+x)}-1&\sim \frac{1}{n}x \end{align*}$

6. Integral formulae

$\begin{align} &\iint\limits_{D}f(x,y)d\sigma = \int_{a}^{b}dx\int_{\varphi_{1}(x)}^{\varphi_{2}(x)}f(x,y)dy=\int_{a}^{b}[\int_{\varphi_{1}(x)}^{\varphi_{2}(x)}f(x,y)dy]dx \\ &\iint\limits_{D}f(x,y)d\sigma = \int_{c}^{d}dy\int_{\psi_{1}(x)}^{\psi_{2}(x)}f(x,y)dx=\int_{c}^{d}[\int_{\psi_{1}(x)}^{\psi_{2}(x)}f(x,y)dx]dy \\ &\iint\limits_{D}f(x,y)dxdy=\iint\limits_{D}f(\rho\cos\theta, \rho\sin\theta)\rho d\rho d\theta \\ &\iint\limits_{D}f(\rho\cos\theta, \rho\sin\theta)\rho d\rho d\theta = \int_{\alpha}^{\beta}d\theta\int_{0}^{\varphi(\theta)}f(\rho\cos\theta, \rho\sin\theta)\rho d\rho \\ &\iint\limits_{D}f(\rho\cos\theta, \rho\sin\theta)\rho d\rho d\theta = \int_{0}^{2\pi}d\theta\int_{0}^{\varphi(\theta)}f(\rho\cos\theta, \rho\sin\theta)\rho d\rho \end{align}$

7. Analytic geometry in three dimensions

Definition 7.1. A surface in $\mathbb{R}^{3}$ is called ${\color{green}{\textbf{a surface of revolution}}}$ if it is generated by rotating a curve around an ${\color{red}{\textrm{axis of rotation}}}$.

Let $C$ be a curve in ${\color{magenta}{YOZ\textrm{-plane}}}$ defined by the equation $f(y,z)=0$. If we rotate $C$ around

1. the ${\color{magenta}{z\textrm{-axis}}}$, then we have $f(\pm\sqrt{x^{2}+y^{2}},z)=0$.
2. the ${\color{magenta}{y\textrm{-axis}}}$, then we have $f(y,\pm\sqrt{x^{2}+z^{2}})=0$.

Example 7.2.  Some examples of ${\color{red}{\textit{quadratic surfaces}}}$ :

1. ${\color{Cyan}{\textbf{Elliptic cone}}} : \displaystyle{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = z^{2}}$.
2. ${\color{green}{\textbf{Ellipsoid}}} : \displaystyle{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1} $.
3. ${\color{magenta}{\textbf{One-sheet hyperboloid}}}$ or ${\color{orange}{\textbf{hyperbolic hyperboloid}}} : \displaystyle{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1}$.
4. ${\color{blue}{\textbf{Two-sheet hyperboloid}}}$ or ${\color{Goldenrod}{\textbf{elliptic hyperboloid}}} : \displaystyle{\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1}$.

8. $\Gamma$ Functions

Definition 8.1. A ${\color{RoyalBlue}{\textbf{$\Gamma$ function}}}$ is a function of the form $$ \Gamma(s) = \int_{0}^{+\infty}e^{-x}x^{s-1}dx  \ (s>0).$$

The $\Gamma$ function has some important properties :

1. $\Gamma(s+1)=s\Gamma(s)$ $(s>0)$.
2. when $s\to0^{+}$, $\Gamma(s)\to +\infty$.
3. $\Gamma(s)\Gamma(1-s)$ = $\displaystyle\frac{\pi}{\sin\pi s}$ $(0<s<1)$.
4. In $\Gamma(s) = \int_{0}^{+\infty}e^{-x}x^{s-1}dx$, if replace $x$ by $u^{2}$, we have $$ \Gamma(s) = 2\int_{0}^{+\infty}e^{-u^{2}}u^{2s-1}du $$ and let $2s-1=t$ or $\displaystyle s=\frac{1+t}{2}$, we have $$ \int_{0}^{+\infty}e^{-u^{2}}u^{t}du = \frac{1}{2}\Gamma(\frac{1+t}{2}) \ (t>-1). $$

9. Applications of definite integration

The following are three cases to solve the ${\color{green}{\textrm{arc length}}}$ of a given curve :

1. Let $C$ be a ${\color{Cyan}{\textrm{parametric curve}}}$ defined by $$ \begin{cases} x=\varphi(t), \ \ (\alpha\leq\beta)\\ y=\psi(t) . \end{cases} $$ Then the ${\color{purple}{\textbf{arc length}}}$ of $C$ is $\displaystyle s=\int_{\alpha}^{\beta}\sqrt{\varphi'^{2}(t)+\psi'^{2}(t)}dt$.
2. If the curve $C$ is defined by the equation $y=f(x)$, $(a\leq x\leq b)$. Then its ${\color{Sepia}{\textbf{arc length}}}$ is $\displaystyle s = \int_{a}^{b}\sqrt{1+y'^{2}}dx$.
3. If the curve $C$ is defined by the polar equation $\rho=\rho(\theta)$, $(\alpha\leq\beta)$. Then its ${\color{RoyalBlue}{\textbf{arc length}}}$ is $\displaystyle s=\int_{\alpha}^{\beta}\sqrt{\rho^{2}(\theta)+\rho'^{2}(\theta)}d\theta.$