• Calculus
• function
  • piecewise defined function
  • composite function $f\circ g$
• mathematical model
  • empirical model
• limit
  • limit law
  • direct substitution property
  • limit comparison
    • squeeze theorem (sandwich theorem)
  • continuity
    • discontinuity
    • intermediate value theorem
• derivative $f^{\prime }(a)$
  • derivative function $f^{\prime }(x)$
  • differentiability
  • second derivative
  • derivative rule
  • implicit differentiation
    • derivative of (inverse) trigonometric function
    • derivative of logarithmic function
  • logarithmic differentiation
  • differential $dy$
  • hyperbolic function
• extremum
  • extreme value theorem
  • Fermat’s theorem
  • critical number $a$
    • closed interval method
  • mean value theorem
    • Rolle’s theorem
  • derivative test
    • first derivative test
    • second derivative test
      • concave upward/ downward
      • inflection point
• indeterminate form
  • L’Hospital’s rule
• antiderivative
• integral
  • definite integral
    • midpoint rule
    • property of definite integral
  • fundamental theorem of calculus
  • indefinite integral
    • net change theorem
  • substitution rule
    • trigonometric integral and trigonometric substitution
  • rational function integral
  • symmetric integral
  • volume from integral
  • arc length from integral
  • surface area from integral
  • average of function
    • mean value theorem for integral
  • integration by part
  • improper integral
    • integral with infinite interval
    • discontinuous integral
    • improper integral comparison

Calculus

function

piecewise defined function

different formula for different part of domain

• step function

composite function $f\circ g$

mathematical model

empirical model

model based entirely on data

limit

$f$ is defined on interval containing $a$ except possibly $a$

$$\begin{gathered} \underset{x\to a}{\lim }f(x)=L \\ \iff \forall \epsilon >0,\exists \delta >0, \\ \text{if}0<|x-a|<\delta \text{then}|f(x)-L|<\epsilon \end{gathered}$$

• limit exist iff limit from both side exist

  $$\underset{x\to a}{\lim }f(x)=L\text{iff }\underset{x\to a^{-}}{\lim }f(x)=L\text{ and }\underset{x\to a^{+}}{\lim }f(x)=L$$

limit law

• linear: sum law, difference law, constant multiplication law

• product law, power law

• quotient law

  $$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}\underset{x\to a}{\lim }g(x)\ne 0$$

derivative rule

• $\forall r>0$ rational

  $$\underset{x\to \infty }{\lim }\frac{1}{x^r}=0$$

  • limit calculation technique
    divide numerator and denominator by highest power of $x$

direct substitution property

polynomial limit can be pulled out

$$\underset{h\to 0}{\lim }\frac{b^h-1}{h}=\ln b$$

limit comparison

$f(x)\le g(x)$ near $a$

$$\Rightarrow \underset{x\to a}{\lim }f(x)\le \underset{x\to a}{\lim }g(x)$$

squeeze theorem (sandwich theorem)

$f(x)\le g(x)\le h(x)$ near $a$,

$$\begin{gathered} \underset{x\to a}{\lim }f(x)=\underset{x\to a}{\lim }h(x)=L \\ \Rightarrow \underset{x\to a}{\lim }g(x)=L \end{gathered}$$

e.g.

$$\underset{h\to 0}{\lim }\frac{\sin h}{h}=1$$

continuity

$f$ is continuous at $a$

$$\iff \underset{x\to a}{\lim }f(x)=f(a)$$

• continuous from the left/ right
• continuous on an interval $\iff$ continuous at every point in interval
• continuous function after operation in limit law are still continuous
• function continuous in domain
  • polynomial
  • rational
  • (inverse) trigonometric
  • exponential
  • logarithmic

discontinuity

• removable
  redefining $f$ at single point remove discontinuity
• unremovable
  • infinite discontinuity
  • jump discontinuity

intermediate value theorem

$f$ continuous on $[a,b],f(a)\ne f(b)$,
$\Rightarrow \forall N\in (f(a),f(b)),\exists c\in (a,b),f(c)=N$

derivative $f^{\prime }(a)$

derivative of $f$ at $a$

$$\begin{gathered} f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h} \\ =\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a} \end{gathered}$$

derivative function $f^{\prime }(x)$

$$\begin{gathered} f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h} \\ =y^{\prime }=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f(x)=Df(x)=D_{x}f(x) \end{gathered}$$

differential operator $D$, $\frac{d}{dx}$

$$f^{\prime }(a)=\frac{dy}{dx}{|}_{x=a}=\frac{dy}{dx}]{}_{x=a}$$

differentiability

$f$ is differentiable at $a$
$\iff f^{\prime }(x)$ exist

$f$ is differentiable on interval
$\iff f$ is differentiable at every point in the interval

• $f$ differentiable at $a\Rightarrow f$ continuous at $a$

second derivative

$$f^{\prime \prime }(x)=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d^{2}y}{d{x}^2}$$

derivative rule

• linear: constant multiplication, sum, difference rule

• definition of $e$

  $$\underset{h\to 0}{\lim }\frac{e^h-1}{h}=1$$

• product rule

  $$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$$

• quotient rule

  $$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$$

• chain rule

  $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$$

implicit differentiation

differentiate both side of equation

derivative of (inverse) trigonometric function

$$\begin{gathered} \frac{d}{dx}(\tan x)={\sec }^{2}x \\ \frac{d}{dx}(\csc x)=-\csc x\cot x \\ \frac{d}{dx}(\sec x)=\sec x\tan x \\ \frac{d}{dx}(\cot x)=-{\csc }^{2}x \\ \frac{d}{dx}({\sin }^{-1}x)=\frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx}({\cos }^{-1}x)=-\frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx}({\tan }^{-1}x)=\frac{1}{1+x^2} \\ \frac{d}{dx}({\cot }^{-1}x)=-\frac{1}{1+x^2} \\ \frac{d}{dx}({\sec }^{-1}x)=\frac{1}{x\sqrt{x^2-1}} \\ \frac{d}{dx}({\csc }^{-1}x)=-\frac{1}{x\sqrt{x^2-1}} \end{gathered}$$

derivative of logarithmic function

$$\begin{gathered} \frac{d}{dx}(\ln |u|)=\frac{du}{udx} \\ \frac{d}{dx}\ln |x|=\frac{1}{x} \end{gathered}$$

logarithmic differentiation

take the logarithm of both side of equation and differentiate

differential $dy$

$$dy=\frac{dy}{dx}dx$$

hyperbolic function

$$\begin{gathered} \sinh x=\frac{e^x-e^{-x}}{2} \\ \cosh x=\frac{e^x+e^{-x}}{2} \end{gathered}$$

• hyperbolic identity similar to trigonometric identity
• inverse hyperbolic function composed of logarithm
• derivative of hyperbolic function compose of hyperbolic functions
• derivative of inverse hyperbolic function have similar form like that of trigonometric function

extremum

extreme value theorem

$f$ continuous on closed interval
$\Rightarrow f$ has absolute maximum and absolute minimum in the interval

Fermat’s theorem

$f$ has extremum at $a$, $f^{\prime }(a)$ exist
$\Rightarrow f^{\prime }(c)=0$

critical number $a$

$f^{\prime }(a)=0$ or $f^{\prime }(a)$ DNE

closed interval method

find absolute extremum of continuous function $f$ on closed interval by checking the value of the critical number and endpoint

mean value theorem

$f$ continuous on $[a,b]$, differentiable on $(a,b)$
$\Rightarrow \exists c\in (a,b),f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

• $\forall x\in (a,b),f^{\prime }(x)=0\Rightarrow \forall x\in (a,b),f(x)=C$
• $\forall x\in (a,b),f^{\prime }(x)=g^{\prime }(x)\Rightarrow \forall x\in (a,b),f(x)=g(x)+C$

Rolle’s theorem

$f$ continuous on $[a,b]$, differentiable on $(a,b)$, $f(a)=f(b)$
$\Rightarrow \exists c\in (a,b),f^{\prime }(c)=0$

derivative test

first derivative test

• increasing/ decreasing
• extremum

second derivative test

concave upward/ downward

graph of $f$ lie above/ below all its tangent

inflection point

indeterminate form

• $\frac{0}{0}$
• $\frac{\infty }{\infty }$

form that can transform to indeterminate form

• $0\cdot \infty$
• $\infty -\infty$
• $0^{\infty }$
• ${\infty }^0$
• $1^{\infty }$

L’Hospital’s rule

$f,g$ differentiable, $g^{\prime }(x)\ne 0$ near $a$ (excluding $a$), ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ is in indeterminate form
$$\Rightarrow \underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

antiderivative

$$F^{\prime }=f$$

• $F(x)+C$ are also antiderivative of $f(x)$

integral

definite integral

$${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x$$

• integrable

• integral sign $\int$

• integrand $f(x)$

• limit of integration

  • lower limit $a$
  • upper limit $b$

• Riemann sum

  $$\sum _{i=1}^{n}f(x_i^{\ast })\Delta x$$

midpoint rule

choose $x^{\ast }$ to be the midpoint

property of definite integral

$$\begin{gathered} {\int }_a^{b}f(x)d(x)=-{\int }_b^{a}f(x)dx \\ {\int }_a^{a}f(x)dx=0 \\ {\int }_a^{b}f(x)d(x)+{\int }_b^{c}f(x)d(x)={\int }_a^{c}f(x)d(x) \\ f(x)\ge 0\Rightarrow {\int }_a^{b}f(x)d(x)\ge 0 \\ f(x)\ge g(x)\Rightarrow {\int }_a^{b}f(x)d(x)\ge {\int }_a^{b}g(x)d(x) \\ m\le f(x)\le M\Rightarrow m(b-a)\le {\int }_a^{b}f(x)d(x)\le M(b-a) \end{gathered}$$

• linear

fundamental theorem of calculus

$f$ continuous on $[a,b]$

$$\Rightarrow g(x)={\int }_a^{x}f(t)dta\le x\le b$$

continuous on $[a,b]$ and differentiable on $(a,b)$
$g^{\prime }(x)=f(x)$

$F$ is any antiderivative of $f$

$$\Rightarrow {\int }_a^{b}f(x)dx=F(b)-F(a)$$

indefinite integral

$$\int f(x)dx$$

mean

$$F^{\prime }(x)=f(x)$$

net change theorem

$${\int }_a^b{F}^{\prime }(x)dx=F(b)-F(a)$$

substitution rule

$$\begin{gathered} \int v(u(x))\frac{du}{dx}dx=\int v(u)du \\ {\int }_a^{b}v(u(x))\frac{du}{dx}dx={\int }_{u(a)}^{u(b)}v(u)du \end{gathered}$$

$\Leftarrow$ chain rule

trigonometric integral and trigonometric substitution

from ${\sin }^{2}x+{\cos }^{2}x=1$

$$\begin{gathered} \sqrt{a-x^2}=\sqrt{a}\cos \theta (x=\sqrt{a}\sin \theta ,\theta \in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]) \\ \int {\sin }^{2k-1}x{\cos }^{n}xdx=-\int (1-{\cos }^{2}x{)}^{k-1}{\cos }^{n}xd(\cos x) \\ \int {\sin }^{m}x{\cos }^{2k-1}xdx=\int {\sin }^{m}x(1-{\sin }^{2}x{)}^{k-1}d(\sin x) \end{gathered}$$

from ${\sec }^{2}x-{\tan }^{2}x=1$

$$\begin{gathered} \sqrt{x^2-a}=\sqrt{a}\tan \theta (x=\sqrt{a}\sec \theta ,\theta \in \left[0,\frac{\pi }{2}\right)\cup \left[\pi ,\frac{3\pi }{2}\right)) \\ \int {\tan }^{m}x{\sec }^{2k}xdx=\int {\tan }^{m}x({\tan }^{2}x+1{)}^{k-1}d(\tan x) \\ \int {\tan }^{2k-1}x{\sec }^{n}xdx=\int ({\sec }^{2}x-1{)}^{k-1}{\sec }^{n-1}xd(\sec x) \end{gathered}$$

$$\begin{gathered} \int \tan xdx=\ln |\sec x|+C \\ \int \sec xdx=\ln |\sec x+\tan x|+C \end{gathered}$$

rational function integral

$$\begin{gathered} \frac{a_1{x}^t+a_2{x}^{t-1}+\cdots +a_{t+1}}{{\left(b_{1}x+c_1\right)}^{m_1}\dots {\left(b_{n}x+c_n\right)}^{m_n}{\left(d_1{x}^2+h_1\right)}^{p_1}\dots {\left(d_q{x}^2+h_q\right)}^{p_q}} \\ =\frac{A_{11}}{b_{1}x+c_1}+\cdots +\frac{A_{1m_1}}{{\left(b_{1}x+c_1\right)}^{m_1}}+\cdots +\frac{A_{n1}}{b_{n}x+c_n}+\cdots +\frac{A_{n{m}_n}}{{\left(b_{n}x+c_n\right)}^{m_n}}+\frac{C_{11}x+D_{11}}{d_1{x}^2+h_1}+\cdots \\ +\frac{C_{1p_1}x+D_{1p_1}}{{\left(d_1{x}^2+h_1\right)}^{p_1}}+\cdots +\frac{C_{q1}x+D_{q1}}{d_q{x}^2+h_q}+\cdots +\frac{C_{q{p}_q}x+D_{q{p}_q}}{{\left(d_q{x}^2+h_q\right)}^{p_q}} \end{gathered}$$

$$\begin{gathered} \int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right) \\ \int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln \left|\frac{x-a}{x+a}\right| \end{gathered}$$

symmetric integral

$$\begin{gathered} f(-x)=f(x)\Rightarrow {\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx \\ f(-x)=-f(x)\Rightarrow {\int }_{-a}^{a}f(x)dx=0 \end{gathered}$$

volume from integral

from $x=a$ to $x=b$, rotate $y=f(x)$ and $y=g(x)$ about $x$-axis, volume:

$${\int }_a^{b}A(x)dx={\int }_a^b\pi \left|(f(x){)}^2-(g(x){)}^2\right|dx$$

from $x=a$ to $x=b$, rotate $y=f(x)$ and $y=g(x)$ about $y$-axis, volume:

$${\int }_a^{b}l(x)h(x)dx={\int }_a^{b}2\pi x|f(x)-g(x)|dx$$

arc length from integral

$$\begin{gathered} {\int }_P^{Q}ds={\int }_P^Q\sqrt{(dx{)}^2+(dy{)}^2} \\ ={\int }_a^b\sqrt{1+{\left(\frac{df}{dx}\right)}^2}dx \end{gathered}$$

surface area from integral

from $x=a$ to $x=b$, rotate $y=f(x)$ about $y$-axis, surface area:

$$\begin{gathered} {\int }_a^{b}l(x)d(arc{ \\ }_{l}ength(x)) \\ ={\int }_a^{b}2\pi f(x)\sqrt{1+{\left(\frac{df}{dx}\right)}^2}dx \end{gathered}$$

average of function

$$f_{ave}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$

mean value theorem for integral

$f$ continuous $[a,b]\Rightarrow \exists c\in [a,b]$

$$f(c)=f_{ave}$$

integration by part

$$\int udv=uv-\int vdu$$

$\Leftarrow$ power rule

improper integral

integral with infinite interval

• $f$ continuous on $[a,\infty )$
• ${\lim }_{t\to \infty }{\int }_a^{t}f(x)dx$ exist

$$\Rightarrow {\int }_a^{\infty }f(x)dx=\underset{t\to \infty }{\lim }{\int }_a^{t}f(x)dx$$

• opposite for ${\int }_{-\infty }^{a}f(x)dx$
• convergent if limit exist
  • divergent if limit DNE
• both side exist $\Rightarrow {\int }_{-\infty }^{\infty }f(x)dx$ is their sum

discontinuous integral

• $f$ continuous on $[a,b)$, discontinuous at $b$
• ${\lim }_{t\to b^{-}}{\int }_a^{t}f(x)dx$ exist

$$\Rightarrow {\int }_a^{b}f(x)dx=\underset{t\to b^{-}}{\lim }{\int }_a^{t}f(x)dx$$

• opposite for when $f$ continuous on $(a,b]$, discontinuous at $a$
• convergent if limit exist
  • divergent if limit DNE
• both side exist $\Rightarrow$ can connect them

improper integral comparison

$f(x)\ge g(x)\ge 0$, ${\int }_a^{\infty }f(x)dx$ converge
$\Rightarrow {\int }_a^{\infty }g(x)dx$ converge