clear

loop = 2000;

v = VideoWriter('sphears2000.avi');

h = figure;

v.FrameRate = 60;

v.Quality = 100;

%disp(v)

open(v);

h.Visible = 'off';

for i = 1:loop

   sphere(i);

   axis equal

   title(i)

   %drawnow

   %disp(getframe)

   disp((i/loop) * 100)

   writeVideo(v, getframe(h));

end

h.Visible = 'on';

close(v);

clear

clc

format long G

startTime = datetime('now');

row = 6.5e4;

%row = 6.506e3;

col = row;

random = rand(row, col);           

disp("Outputing")

%writematrix(random,"Good_Luck2.xlsx")

sum = sum(random, 'all');

disp(sum)

count = row * col;

disp(count)

average = mean(random, 'all');

disp(average)

endTime = datetime('now');

totalTime = between(startTime, endTime);

disp(totalTime)

clear

clc

clf

format long G

iterations = 3000;

digits(5000)

% greg's infinite series

denomenator = 1;

gregPi = vpa(0);

%Nilakantha's infinite series

num = 2;

nilPi = vpa(3);

%ran pi

coPrime = 0;

ranPi = 0;

randMax = 2^53 - 1;

%formula using arctan

arc5 = 0;

odd = 1;

number5 = vpa(((1/5)^odd) / odd);

arc239 = 0;

number239 = vpa(((1/239)^odd) / odd);

for i = 1:iterations

   % greg's infinite series

   if mod(i, 2) == 0

       gregPi = gregPi + (4/denomenator);

   else

       gregPi = gregPi - (4/denomenator);

   end

   gregy(i) = percentError(abs(gregPi));

   denomenator = denomenator + 2;

   %formula using arctan

   if(mod(i, 2) == 0)

           arc5 = arc5 - number5;

           arc239 = arc239 - number239;

       else   

           arc5 = arc5 + number5;

           arc239 = arc239 + number239;

   end

       odd = odd + 2;

       number5 = ((1/5)^odd) / odd;

       number239 = ((1/239)^odd) / odd;

   x(i) = i;

   arcy(i) = percentError(vpa(4*((4*abs(arc5))- abs(arc239))));

   piey(i) = [vpa(pi)];

   %Nilakantha's infinite series

   if mod(i - 1, 2) == 0

       nilPi = nilPi + (vpa(4/(num*(num+1)*(num+2))));

   else

       nilPi = nilPi - (vpa(4/(num*(num+1)*(num+2))));

   end

   nily(i) = percentError(nilPi);

   num = num + 2;

   %ran pi

   if(iscoprime([randi([1, randMax]), randi([1,randMax])]))

       coPrime = coPrime + 1;

   end

   if(coPrime > 0)

       rany(i) = percentError(sqrt(6/(coPrime/(i + 1))));

   end

end

hold on

start = 1;

%plot(x, piey, 'LineWidth', 2)

plot(x(start:end), arcy(start:end))

plot(x(start:end), gregy(start:end))

plot(x(start:end), nily(start:end))

plot(x(start:end), rany(start:end))

legend('ArcTan', 'Greg', 'Nil', 'Random')

%ylim([0, 0.5])

disp('Done')

function y = percentError(num)

   y = abs((num - vpa(pi)) / vpa(pi)) * 100;

end

clear all

close all

clc

x = [8 10];

y = [2 -2];

n = length(x);

P = "";

for i = 1:n

   P = P + "((" + y(i) + "*";

   for j = 1:n

       if(j ~= i)

           if (x(j) > 0)

            P = P + "(x-" + x(j) + ")*";

           else

               P = P + "(x+" + abs(x(j)) + ")*";

           end

       end

   end

   P = extractBefore(P, strlength(P));

   P = P + ")/";

   temp = 1;

   for j = 1:n

       if(j ~= i)

           temp = temp * (x(i) - x(j));

       end

   end

   P = P + temp + ")" + "+";

end

P = extractBefore(P, strlength(P));

disp(P);

\documentclass[tikz, 12pt]{article}

\usepackage[margin=0.7in]{geometry}

\usepackage{amsmath}

\usepackage[makeroom]{cancel}

\usepackage{graphicx}

\usepackage[utf8]{inputenc}

\usepackage{ stmaryrd }

\usepackage{multicol}

\usepackage{tasks}

\usepackage{tabularx}

\usepackage{tasks}

\usepackage{vwcol}

\usepackage{booktabs}

\usepackage{tabularray}

\usepackage{pgfplots}

\usepackage{fancyhdr}

\usepackage{xcolor}

\usepackage{soul}

\usepgflibrary{plotmarks}

\def\newline{\hfill \break}

\def\({\left(}

\def\){\right)}

\def\ddx{\frac{d}{dx}}

\def\[{\left[}

\def\]{\right]}

\def\ddy{\frac{dy}{dx}}

\setul{0.5ex}{0.2ex}

\setulcolor{red}

\begin{document}

\begin{flushleft}

\begin{large}\textbf{Section 4.1: Minimum and Maximum Values (Part a)}\end{large} \\

\newline

In this section we will learn how to use the derivative of a function to find its maximum and

minimum values. \\

Furthermore, we will learn conditions which guarantee that a function has extreme values, and

methods for determining their location. \\

\newline

\textbf{Warm Up} \\

\newline

Suppose the graph of a function $f$ below. What is the \ul{highest} and \ul{lowest} values of $f$. \\

\newline

\begin{center}

\begin{tikzpicture}

\begin{axis} [

width = 0.5\textwidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 6,

xmin = -4.5, xmax = 5.5,

xtick={-4, -3, -2, -1, 0, 1, 2, 3, 4, 5

},

ytick = {-1, 0, 1, 2, 3, 4, 5},

xlabel = {$x$},

ylabel = {$y$},

every axis plot/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

]

\addplot [

black,

samples = 200,

domain = -3:1,

]

{((5*(x+3)*(x+0)*(x-1))/4)+((2*(x+1)*(x+0)*(x-1))/-24)+((4*(x+1)*(x+3)*(x-1))/-3)+((1*(x+1)*(x+3)*(x+0))/8)};

\addplot [

blue,

samples = 200,

domain = -1.5:-.5,

]

{5};

\addplot [

black,

samples = 200,

domain = 1:4,

]

{((1*(x-1.5)*(x-2.25)*(x-4))/-1.875)+((2*(x-1)*(x-2.25)*(x-4))/0.9375)+((3*(x-1)*(x-1.5)*(x-4))/-1.6406)+((4*(x-1)*(x-1.5)*(x-2.25))/13.125)} node[right]{$y = f(x)$};

\addplot [red, dotted] coordinates {(-3, 0) (-3, 2)};

\addplot [red, samples = 200, domain = -3:0, dotted]{2};

\addplot [red, dotted] coordinates {(-1, 0) (-1, 5)};

\addplot [red, samples = 200, domain = -1:0, dotted]{5};

\addplot [red, dotted] coordinates {(1, 0) (1, 1)};

\addplot [red, samples = 200, domain = 0:1, dotted]{1};

\addplot [red, dotted] coordinates {(4, 0) (4, 4)};

\addplot [red, samples = 200, domain = 0:4, dotted]{4};

\end{axis}

\end{tikzpicture}

\end{center}

\begin{flalign*}

\text{The Highest} &= \text{Largest Value} & f(-1) &= 5 \text{ The largest value of } f\\

\text{The Lowest} &= \text{Smallest Value} & f(1) &= 1 \text{ The smallest value of } f

\end{flalign*}

$$\text{Domain }f\text{:} [-3,4]$$ \\

\begin{flalign*}

\text{The maximum value is } 5 \text{ at } x &= -1 \\

\text{The minimum value is } 1 \text{ at } x &= 1

\end{flalign*}

\fbox{

\parbox{\textwidth}{

\newline

\textbf{Definition} \\

\newline

Let \ul{$c$} be a number \ul{in the domain $D$} of a function $f$.

\begin{itemize}

\item{If $f(c) \ge f(x)$ for all $x$ in $D$, then $f(c)$ is an \textbf{\ul{absolute maximum}} (or \textbf{\ul{global maximum}}) value.}

\item{If \color{red}\fbox{\color{black}$f(c) \le f(x)$} \color{black} for all $x$ in $D$, then $f(c)$ is an \textbf{\ul{absolute minimum}} (or \textbf{\ul{global minimum}}) value.}

\end{itemize}

\newline

The minimum and maximum values of $f$ are known as \textbf{\ul{extreme values}} or just \textbf{extrema}.

}}

\clearpage

How do we find absolute extrema? \\

\vspace{4ex}

\begin{tblr}{

width=\textwidth,

rows = {halign = c, valign = m}, colspec={X X},}

{\begin{tikzpicture}

\begin{axis} [

width = \linewidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 3,

xmin = -2, xmax = 2,

xtick={-1, 0, 1},

ytick = {0, 1, 2},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

title = {$f(x) = x^2 + 1 \text{ on } \(-\infty, \infty\)$},

]

\addplot [

black,

samples = 200,

domain = -1:1,

style={<->}

]

{x^2 + 1};

\addplot[mark=*] coordinates {(0,1)};

\addplot [

red,

dotted,

samples = 200,

domain = -0.5:0.5,

]

{1} node[right]{$f'(0) = 0$};

\node[red, thick] (source) at (axis cs:-1.5,0.5){abs min is $1$ at $x=0$};

\node (destination) at (axis cs:0, 1){};

\draw[red, thick, ->](source)--(destination);

\end{axis}

\end{tikzpicture} \\

There is no abs max} &

{\begin{tikzpicture}

\begin{axis} [

width = \linewidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 3,

xmin = -2, xmax = 2,

xtick={-1, 0, 1},

ytick = {0, 1, 2},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

title = {$f(x) = x^2 + 1 \text{ on } \[0, 1\]$},

]

\addplot [

black,

samples = 200,

domain = 0:1,

]

{x^2 + 1};

\addplot[mark=*] coordinates {(0,1)};

\addplot[mark=*] coordinates {(1,2)};

\addplot [

red,

dotted,

samples = 200,

domain = 0:1,

]

{2};

\node[red, thick] (source) at (axis cs:-1.5,0.5){abs min is $1$ at $x=0$};

\node (destination) at (axis cs:0, 1){};

\draw[red, thick, ->](source)--(destination);

\node[red, thick] (source) at (axis cs:2,3){abs max is $2$ at $x=1$};

\node (destination) at (axis cs:1, 2){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(1, 0) (1, 2)};

\end{axis}

\end{tikzpicture} \\

The abs max and min occur at the endpoints} \\

\end{tblr} \\

\vspace{10ex}

\begin{tblr}{

width=\textwidth,

rows = {halign = c, valign = m}, colspec={X X}}

{\begin{tikzpicture}

\begin{axis} [

width = \linewidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 3,

xmin = -2, xmax = 2,

xtick={-1, 0, 1},

ytick = {0, 1, 2},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

title = {$f(x) = x^2 + 1 \text{ on } \( 0, 1\]$},

]

\addplot [

black,

samples = 200,

domain = 0:1

]

{x^2 + 1};

\fill[white,draw=black, very thick](axis cs:0,1)circle[radius=3pt];

\addplot[mark=*] coordinates {(1,2)};

\addplot [

red,

dotted,

samples = 200,

domain = 0:1,

]

{2};

\node[red, thick] (source) at (axis cs:-1.5,0.5){No abs min};

\node (destination) at (axis cs:0, 1){};

\draw[red, thick, ->](source)--(destination);

\node[red, thick] (source) at (axis cs:2,3){abs max is $2$ at $x=1$};

\node (destination) at (axis cs:1, 2){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(1, 0) (1, 2)};

\end{axis}

\end{tikzpicture}} &

{\begin{tikzpicture}

\begin{axis} [

width = \linewidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 3,

xmin = -2, xmax = 2,

xtick={-1, 0, 1},

ytick = {0, 1, 2},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

title = {$f(x) = x^2 + 1 \text{ on } \(0, 1\)$},

]

\addplot [

black,

samples = 200,

domain = 0:1,

]

{x^2 + 1};

\fill[white,draw=black, very thick](axis cs:0,1)circle[radius=3pt];

\addplot [

red,

dotted,

samples = 200,

domain = 0:1,

]

{2};

\node[red, thick] (source) at (axis cs:-1.5,0.5){No abs min};

\node (destination) at (axis cs:0, 1){};

\draw[red, thick, ->](source)--(destination);

\node[red, thick] (source) at (axis cs:2,3){No abs max};

\node (destination) at (axis cs:1, 2){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(1, 0) (1, 2)};

\fill[white,draw=black, very thick](axis cs:1,2)circle[radius=3pt];

\end{axis}

\end{tikzpicture}} \\

\end{tblr}

\clearpage

\fbox{

\parbox{\textwidth}{

\newline

\textbf{Definition} \\

\newline

Let \ul{$c$ be a number} in the domain $D$ of a function $f$.

\begin{itemize}

\item{The number $f(c)$ is a \textbf{\ul{local minimum}} (or \textbf{\ul{relative minimum}}) value of $f$ if \\

\color{red}\fbox{\color{black}$f(c) \le f(x)$} \color{black} for \ul{all $x$} in an \ul{open interval containing $c$}}

\item{The number $f(c)$ is a \textbf{\ul{local maximum}} (or \textbf{\ul{relative maximum}}) value of $f$ if \\

\color{red}\fbox{\color{black}$f(c) \ge f(x)$} \color{black} for \ul{all $x$} in an \ul{open interval containing $c$}}

\end{itemize}

Local maxima/minima are sometimes referred to as \textbf{local extrema}.

}} \\

\newline

\textbf{Note:} Following along with Stewart’s convention, \ul{local extrema will only occur on the

\textit{interior} of an interval}, and \ul{never at the endpoints}. However, \ul{endpoints can be absolute

extrema}. \\

\newline

How do we find local extrema? \\

\newline

The $x$ value where $f'(c) = 0$ or $f'(c)$ Does not exist (corner, cusp) \\

\newline

\textbf{Example 1} \\

\newline

Use the graph to state the \ul{absolute} and \ul{local maximum and minimum} values of the

function. \\

\end{flushleft}

\begin{center}

\newline

\begin{tikzpicture}

\begin{axis} [

width = 0.9\textwidth,

axis x line=middle,

axis y line=middle,

ymin = -4.5, ymax = 6.5,

xmin = -7.5, xmax = 7.5,

xtick={-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7},

ytick = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick}

]

\addplot[mark=*] coordinates {(0,0)};

\addplot [

black,

samples = 200,

domain = -7:-6,

]

{((-3*(x+6))/-1)+((1*(x+7))/1)};

\node[red, thick] (source) at (axis cs:-8,-4){\fbox{abs min}};

\node (destination) at (axis cs:-7, -3){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(-7, 0) (-7, -3)};

\addplot[red, dotted, domain = -7:0]{-3};

\node[red, thick] (source) at (axis cs:-7,2){Loc max};

\node (destination) at (axis cs:-6, 1){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(-6, 0) (-6, 1)};

\addplot[red, dotted, domain = -6:0]{1};

\addplot [

black,

samples = 200,

domain = -6:-5,

]

{((-2*(x+6)*(x+5.5))/0.5)+((1*(x+5)*(x+5.5))/0.5)+((-1*(x+5)*(x+6))/-0.25)};

\node[red, thick] (source) at (axis cs:-4,-2.5){Loc min};

\node (destination) at (axis cs:-5, -2){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(-5, 0) (-5, -2)};

\addplot[red, dotted, domain = -5:0]{-2};

\node[red, thick] (source) at (axis cs:-5.5, 3){Loc max};

\node (destination) at (axis cs:-4.5, 2){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(-4.5, 0) (-4.5, 2)};

\addplot[red, dotted, domain = -4.5:0]{2};

\addplot [

black,

samples = 200,

domain = -5:-4,

]

{((-2*(x+4.5)*(x+4)*(x+4.25))/-0.375)+((2*(x+5)*(x+4)*(x+4.25))/0.0625)+((1*(x+5)*(x+4.5)*(x+4.25))/0.125)+((1.5*(x+5)*(x+4.5)*(x+4))/-0.046875)};

\node[red, thick] (source) at (axis cs:-3, -1){Loc min};

\node (destination) at (axis cs:-4, 1){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(-4, 0) (-4, 1)};

\addplot [

black,

samples = 200,

domain = -4:-2,

]

{((1*(x+3.5)*(x+3)*(x+2)*(x+2.25))/1.75)+((3*(x+4)*(x+3)*(x+2)*(x+2.25))/-0.46875)+((3.5*(x+4)*(x+3.5)*(x+2)*(x+2.25))/0.375)+((2.5*(x+4)*(x+3.5)*(x+3)*(x+2.25))/0.75)+((3*(x+4)*(x+3.5)*(x+3)*(x+2))/-0.41016)};

\node[red, thick] (source) at (axis cs:-4, 4.5){Loc max};

\node (destination) at (axis cs:-3, 3.5){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(-3, 0) (-3, 3.5)};

\addplot[red, dotted, domain = -3:0]{3.5};

\addplot [

black,

samples = 200,

domain = -2:-1,

]

{((2.5*(x+1)*(x+1.75))/0.25)+((1*(x+2)*(x+1.75))/0.75)+((2*(x+2)*(x+1))/-0.1875)};

\addplot [

black,

samples = 200,

domain = -1:-0,

]

{((1*(x+0))/-1)+((0*(x+1))/1)};

\addplot [

black,

samples = 200,

domain = 0:2,

]

{((5*(x+0)*(x-1))/2)+((0*(x-2)*(x-1))/2)+((3.75*(x-2)*(x+0))/-1)};

\node[red, thick] (source) at (axis cs:3, 6){\fbox{Loc max and abs max}};

\node (destination) at (axis cs:2, 5){};

\draw[red, thick, ->](source)--(destination);

\addplot [red, dotted] coordinates {(2, 0) (2, 5)};

\addplot[red, dotted, domain = 0:2]{5};

\addplot [

black,

samples = 200,

domain = 2:5,

]

{((5*(x-2.5)*(x-4)*(x-4.75)*(x-5))/8.25)+((4*(x-2)*(x-4)*(x-4.75)*(x-5))/-4.2188)+((3*(x-2)*(x-2.5)*(x-4.75)*(x-5))/2.25)+((2.5*(x-2)*(x-2.5)*(x-4)*(x-5))/-1.1602)+((2*(x-2)*(x-2.5)*(x-4)*(x-4.75))/1.875)};

\node[black, thick] at (axis cs:6,3){\large$y=f(x)$};

\addplot [red, dotted] coordinates {(6, 0) (6, 0.5)};

\addplot[red, dotted, domain = 0:6]{0.5};

\addplot [

black,

samples = 200,

domain = 5:6,

]

{((0.5*(x-5))/1)+((2*(x-6))/-1)};

\end{axis}

\end{tikzpicture}

\end{center}

\pagebreak

\begin{flushleft}

\textbf{Example 2} \\

\newline

For each of the following, determine whether \ul{such a function exists}. If so, provide an example.

\begin{enumerate}

\item{ A \ul{continuous function} defined for \ul{all real numbers} \ul{with no absolute extrema}. That is, a continuous function with no absolute maximum and no absolute minimum. \\

\newline

\begin{tblr}{

width=\textwidth,

rows = {halign = c, valign = m}, colspec={X Q[c, m] X}}

{\begin{tikzpicture}

\begin{axis} [

width = 0.5\linewidth,

ticks=none,

axis x line=middle,

axis y line=middle,

ymin = -8, ymax = 8,

xmin = -2, xmax = 2,

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick, <->},

clip = false,

axis line style = {<->, thick},

title = {$f(x) = x^3$},

]

\addplot [

black,

samples = 200,

domain = -2:2

]

{x^3};

\end{axis}

\end{tikzpicture}} & or &

{\begin{tikzpicture}

\begin{axis} [

width = 0.5\linewidth,

ticks=none,

axis x line=middle,

axis y line=middle,

ymin = -2, ymax = 2,

xmin = -2, xmax = 2,

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick, <->},

clip = false,

axis line style = {<->, thick},

title = {$f(x) = x$},

]

\addplot [

black,

samples = 200,

domain = -2:2,

]

{x};

\end{axis}

\end{tikzpicture}} \\

\end{tblr} \begin{center} No abs max and no abs min \end{center}}

\item{A \ul{function} defined on the closed \ul{interval $[0, 1]$} with no \ul{absolute maximum} and \ul{no absolute minimum}. \\

\begin{center}

\begin{tikzpicture}

\begin{axis} [

width = 0.25\linewidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 1,

xmin = -1, xmax = 1,

xtick = {0, 1},

ytick = {1/2, 1},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

title = {{$\displaystyle{f(x)=\begin{cases} \frac{1}{2}, & x=0, x=1 \\ x, & 0<x<1\end{cases}}$}},

]

\addplot [

black,

samples = 200,

domain = 0:1,

]

{x};

\fill[white,draw=black, very thick](axis cs:0,0)circle[radius=3pt];

\fill[white,draw=black, very thick](axis cs:1,1)circle[radius=3pt];

\addplot[mark=*] coordinates {(0,1/2)};

\addplot[mark=*] coordinates {(1,1/2)};

\end{axis}

\end{tikzpicture} \\

No abs max and no abs min \end{center}}

\item{ A \ul{\textbf{continuous} function} defined on the \ul{closed interval $[0, 1]$} with no \ul{absolute maximum} and \ul{no absolute minimum}. \\

\newline

Impossible to find continuous function defined in close interval without abs max and abs min \\

\begin{center}

\begin{tikzpicture}

\begin{axis} [

width = 0.25\linewidth,

axis x line=middle,

axis y line=middle,

ymin = -1, ymax = 3,

xmin = -1, xmax = 7*pi / 2,

xtick = {pi/2, 7*pi/2},

xlabel = {$x$},

ylabel = {$y$},

every axis plot post/.append style={ultra thick},

clip = false,

axis line style = {<->, thick},

xticklabels={$a$, $b$}

]

\addplot [

black,

samples = 200,

domain = pi/2:7*pi / 2,

]

{cos(deg(x+pi/2))+2};

\addplot [red, dotted] coordinates {(pi/2, 0) (pi/2, 1)};

\addplot[red, dotted, domain = 0:pi/2]{1};

\addplot [red, dotted] coordinates {(7 *pi/2, 0) (7 *pi/2, 3)};

\addplot[red, dotted, domain = 0:7 * pi/2]{3};

\addplot[blue, domain = 3 * pi/2 - 1:3 * pi/2 + 1]{3};

\addplot[blue, domain = 5 * pi/2 - 1:5 * pi/2 + 1]{1};

\end{axis}

\end{tikzpicture}

\end{center}

}

\end{enumerate}

\fbox{

\parbox{\textwidth}{

\newline

\textbf{The Extreme Value Theorem} \\

\newline

If $f$ is \ul{continuous} on a closed interval $[a, b]$, then $f$ attains an \ul{absolute maximum value

$f(c)$} and an \ul{absolute minimum value $f(d)$} at\ul{ some numbers $c$ and $d$ in $[a, b]$}.\\}} \\

\newline

\textbf{Note: } The EVT does \textbf{NOT} say where these values are located.

\end{flushleft}

\end{document}

clear

clc

n = 10^6;

sum = 0

for i = 1:n

    sum = sum + sums(i, n);

end

disp(sum)

function x = sums(i, n)

    x = (5/n)*sqrt(1+((5*i)/n));

end

clear

clc

clf

close all

format long

%Rieman Sum Variables

a = 1;

b = exp(1);

n = 100;

x = linspace(a,b,100);

fun = @(x)  log(x);

y = fun(x);

h = figure;

exact = integral(fun, a, b, 'RelTol',0,'AbsTol', 1e-12);

%Video Variables

name = func2str(fun);

writer = VideoWriter(name +  " " + n + '.avi');

writer.FrameRate = floor(n / 10);

writer.Quality = 100;

open(writer);

writeVideo(writer, getframe(h));

hold on

for i = 1:n

   clf

   plot(x,y, 'LineWidth', 1);

   sum = 0;

   for j = 1:i

       title(name + " Rectangles: " + i + " Estimation: " + sum + " Exact: " + exact);

       deltaX = (b - a) / i;

       x2 = a + j*deltaX;

       x1 = x2 - deltaX;

       height = fun(x1);

       if (height < 0)

           r = rectangle('Position',[x1, height, deltaX, abs(height)], ...

               'FaceColor','b');

       else

           r = rectangle('Position',[x1, 0, deltaX, height], ...

               'FaceColor','r');

       end

       if (i >= 500)

           r.LineStyle = 'none';

       end

       sum = sum + (deltaX * height);

       uistack(r,'bottom')

   end

   writeVideo(writer, getframe(h));

   disp ((i / n) * 100);

end

close(writer);

disp('Done');

clear

clc

clf

close all

format long

func = @(x) (1+x.^2);

P = [1 1.5 2 3 4 5];

c = [1 1.5 2 3 4];

d = [1.5 2 3 4 5];

a = P(1);

b = P(length(P));

x = linspace(a,b,1000);

y = func(x);

lineWidth = 10;

plot(x, y, "Color", "black", "LineWidth", lineWidth);

hold on

axis = gca;

axis.FontSize = 50;

%for i = 1:length(c)

% plot([c(i) c(i)], [0 func(c(i))], ...

% "Color", "red", "LineWidth", lineWidth);

%end

sum = 0;

for i=1:length(P) - 1

x1 = P(i);

x2 = P(i + 1);

deltaX = x2 - x1;

height = func(c(i));

sum = sum + (deltaX * height);

r = rectangle('Position',[x1, 0, deltaX, abs(height)], ...

'FaceColor','b');

uistack(r,'bottom')

end

disp(sum);

sum = 0;

for i=1:length(P) - 1

x1 = P(i);

x2 = P(i + 1);

deltaX = x2 - x1;

height = func(d(i));

sum = sum + (deltaX * height);

r = rectangle('Position',[x1, 0, deltaX, abs(height)], ...

'FaceColor','r');

uistack(r,'bottom')

end

disp(sum);

clear

clc

clf

close all

format longg

syms f(x)

f(x) = ((75*x) - x^3 - 125);

dxfunc = diff(f, x);

x_n = 2;

n = 6;

for i = 1:n

num = f(x_n);

den = dxfunc(x_n);

temp = x_n;

x_n = vpa(x_n - (num / den));

if(temp == x_n)

disp("Lost Percision at n = " + i)

break;

end

end

disp(x_n)

clear

clc

clf

close all

format long

syms f(x)

f(x) = 8 - x^2;

a = 0;

b = sqrt(8);

h = b;

x = linspace(a,b,1000);

y = f(x);

hold on;

view(3)

plot3(zeros(length(x), 1), x, y, 'r');

plot3(y, x, zeros(length(x), 1), 'b')

set(gca, 'ydir', 'reverse')

set(gca, 'xdir', 'reverse')

xlabel('x')

ylabel('y')

zlabel('z')

p1 = [0 h/2 f(h/2)];

p2 = [0 h/2 0];

p3 = [f(h/2) h/2 0];

p4 = [f(h/2) h/2 f(h/2)];

rectangle = fill3([p1(1) p2(1) p3(1) p4(1) p1(1)], [p1(2) p2(2) p3(2) p4(2) p1(2)], ...

[p1(3) p2(3) p3(3) p4(3) p1(3)], 'y');

view(-75,26)

clear

clc

clf

close all

format long

syms f(x)

f(x) = sqrt(1 + cos(x)^2);

a = 0;

b = pi / 4;

n = 100000;

deltaX = (b - a) / n;

sum = 0;

alt = 4;

for i = 1:(n + 1)

x = vpa(f(a + (deltaX * (i - 1))));

if(i == 1 || i == (n + 1))

alt = 1;

end

sum = sum + (alt * x);

if (alt == 4)

alt = 2;

else

alt = 4;

end

end

disp(vpa((deltaX / 3) * sum));

tic

clear

clc

clf

close all

format long

syms f(x)

f(x) = sqrt(1 + cos(x)^2);

func = @(x) sqrt(1 + cos(x).^2);

a = 0;

b = pi / 4;

n = 1000000;

deltaX = (b - a) / n;

tic

y = vpa(f(linspace(a + deltaX, b - deltaX, n - 1)));

disp("Got y")

toc

total = f(a) + f(b);

tic

odd = 4 .* y(1:2:end);

even = 2 .* y(2:2:end);

disp("Got odds and evens")

toc

tic

total = total + sum(odd) + sum(even);

disp("Got Total")

toc

disp(vpa((deltaX / 3) * total));

disp(vpa(integral(func, a, b)))

disp("Done")

toc

clear

clc

clf

close all

format long

syms f(x)

f(x) = sqrt(1 + cos(x)^2);

a = 0;

b = pi / 4;

n = 1000;

x = 1:1:n;

version1Total = [];

version2Total = [];

for i = 1:n

version1Function = @() version1(f, a, b, i);

version1Total = [version1Total timeit(version1Function)];

version2Function = @() version2(f, a, b, i);

version2Total = [version2Total timeit(version2Function)];

end

hold on

scatter(x, version1Total, 'b', 'filled')

scatter(x, version2Total, 'r', 'filled')

xlabel('Value of n')

ylabel('Time (s)')

xFit = linspace(min(x), max(x), 1000);

coefficients1 = polyfit(x, version1Total, 1);

yFit1 = polyval(coefficients1 , xFit);

coefficients2 = polyfit(x, version2Total, 1);

yFit2 = polyval(coefficients2 , xFit);

plot(xFit, yFit1, 'b');

plot(xFit, yFit2, 'r');

version1Legend = coefficients1(1) + "x +" + coefficients1(2);

version2Legend = coefficients2(1) + "x +" + coefficients2(2);

legend("Version 1", "Version 2", version1Legend, version2Legend)

function h = version1(f, a, b, n)

deltaX = (b - a) / n;

sum = 0;

alt = 4;

for i = 1:(n + 1)

x = vpa(f(a + (deltaX * (i - 1))));

if(i == 1 || i == (n + 1))

alt = 1;

end

sum = sum + (alt * x);

if (alt == 4)

alt = 2;

else

alt = 4;

end

end

h = vpa((deltaX / 3) * sum);

end

function h = version2(f, a, b, n)

deltaX = (b - a) / n;

y = vpa(f(linspace(a + deltaX, b - deltaX, n - 1)));

total = f(a) + f(b);

odd = 4 .* y(1:2:end);

even = 2 .* y(2:2:end);

total = total + sum(odd) + sum(even);

h = vpa((deltaX / 3) * total);

end

clear

clc

clf

close all

format longg

decimals = 14;

digits(decimals)

syms a_n(x)

a_n(x) = ((-1)^(x+1) * ((1+sqrt(3*x)) / (x + 2)));

n = 1;

max = 10^6;

x = a_n(linspace(n, max, max));

sum = sum(x);

disp(vpa(sum))

disp(0.43554155527698)