Generate Math Exams and Quizzes in LaTeX and pdf formats

Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{21 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 8}{- \frac{63 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 5\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (5.0000000000) - \frac{- \frac{21 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 8}{- \frac{63 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 4.5850390580\end{equation*} \begin{equation*}x_{2} = (4.5850390580) - \frac{- \frac{21 (4.5850390580)^{3}}{250} + \cos{\left((4.5850390580) \right)} + 8}{- \frac{63 (4.5850390580)^{2}}{250} - \sin{\left((4.5850390580) \right)}} = 4.5330828135\end{equation*} \begin{equation*}x_{3} = (4.5330828135) - \frac{- \frac{21 (4.5330828135)^{3}}{250} + \cos{\left((4.5330828135) \right)} + 8}{- \frac{63 (4.5330828135)^{2}}{250} - \sin{\left((4.5330828135) \right)}} = 4.5323883830\end{equation*} \begin{equation*}x_{4} = (4.5323883830) - \frac{- \frac{21 (4.5323883830)^{3}}{250} + \cos{\left((4.5323883830) \right)} + 8}{- \frac{63 (4.5323883830)^{2}}{250} - \sin{\left((4.5323883830) \right)}} = 4.5323882619\end{equation*} \begin{equation*}x_{5} = (4.5323882619) - \frac{- \frac{21 (4.5323882619)^{3}}{250} + \cos{\left((4.5323882619) \right)} + 8}{- \frac{63 (4.5323882619)^{2}}{250} - \sin{\left((4.5323882619) \right)}} = 4.5323882619\end{equation*}

Download \(\LaTeX\)

\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\cos{\left(x \right)}= \frac{21 x^{3}}{250} - 8$ using $x_0=5$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{21 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 8}{- \frac{63 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}}  \end{equation*}
Using $x_0 = 5$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} =  (5.0000000000) - \frac{- \frac{21 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 8}{- \frac{63 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 4.5850390580\end{equation*}
\begin{equation*}x_{2} =  (4.5850390580) - \frac{- \frac{21 (4.5850390580)^{3}}{250} + \cos{\left((4.5850390580) \right)} + 8}{- \frac{63 (4.5850390580)^{2}}{250} - \sin{\left((4.5850390580) \right)}} = 4.5330828135\end{equation*}
\begin{equation*}x_{3} =  (4.5330828135) - \frac{- \frac{21 (4.5330828135)^{3}}{250} + \cos{\left((4.5330828135) \right)} + 8}{- \frac{63 (4.5330828135)^{2}}{250} - \sin{\left((4.5330828135) \right)}} = 4.5323883830\end{equation*}
\begin{equation*}x_{4} =  (4.5323883830) - \frac{- \frac{21 (4.5323883830)^{3}}{250} + \cos{\left((4.5323883830) \right)} + 8}{- \frac{63 (4.5323883830)^{2}}{250} - \sin{\left((4.5323883830) \right)}} = 4.5323882619\end{equation*}
\begin{equation*}x_{5} =  (4.5323882619) - \frac{- \frac{21 (4.5323882619)^{3}}{250} + \cos{\left((4.5323882619) \right)} + 8}{- \frac{63 (4.5323882619)^{2}}{250} - \sin{\left((4.5323882619) \right)}} = 4.5323882619\end{equation*}
}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Use Newton's method to find the first 5 approximations of the solution to the equation  <img class="equation_image" title=" \displaystyle \cos{\left(x \right)}= \frac{21 x^{3}}{250} - 8 " src="/equation_images/%20%5Cdisplaystyle%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B21%20x%5E%7B3%7D%7D%7B250%7D%20-%208%20" alt="LaTeX:  \displaystyle \cos{\left(x \right)}= \frac{21 x^{3}}{250} - 8 " data-equation-content=" \displaystyle \cos{\left(x \right)}= \frac{21 x^{3}}{250} - 8 " />  using  <img class="equation_image" title=" \displaystyle x_0=5 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D5%20" alt="LaTeX:  \displaystyle x_0=5 " data-equation-content=" \displaystyle x_0=5 " /> . </p> </p>

HTML for Canvas

<p> <p>Using the formula for Newton's method gives
 <img class="equation_image" title=" x_{n+1} =  x_{n} - \frac{- \frac{21 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 8}{- \frac{63 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B21%20x_%7Bn%7D%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B63%20x_%7Bn%7D%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%7D%20%20%20" alt="LaTeX:  x_{n+1} =  x_{n} - \frac{- \frac{21 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 8}{- \frac{63 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{21 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 8}{- \frac{63 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}}   " /> 
Using  <img class="equation_image" title=" \displaystyle x_0 = 5 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%205%20" alt="LaTeX:  \displaystyle x_0 = 5 " data-equation-content=" \displaystyle x_0 = 5 " />  and  <img class="equation_image" title=" \displaystyle n = 0,1,2,3, " src="/equation_images/%20%5Cdisplaystyle%20n%20%3D%200%2C1%2C2%2C3%2C%20" alt="LaTeX:  \displaystyle n = 0,1,2,3, " data-equation-content=" \displaystyle n = 0,1,2,3, " />  and  <img class="equation_image" title=" \displaystyle 4 " src="/equation_images/%20%5Cdisplaystyle%204%20" alt="LaTeX:  \displaystyle 4 " data-equation-content=" \displaystyle 4 " />  gives:
 <img class="equation_image" title=" x_{1} =  (5.0000000000) - \frac{- \frac{21 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 8}{- \frac{63 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 4.5850390580 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%285.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B21%20%285.0000000000%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B63%20%285.0000000000%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%7D%20%3D%204.5850390580%20" alt="LaTeX:  x_{1} =  (5.0000000000) - \frac{- \frac{21 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 8}{- \frac{63 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 4.5850390580 " data-equation-content=" x_{1} =  (5.0000000000) - \frac{- \frac{21 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 8}{- \frac{63 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 4.5850390580 " /> 
 <img class="equation_image" title=" x_{2} =  (4.5850390580) - \frac{- \frac{21 (4.5850390580)^{3}}{250} + \cos{\left((4.5850390580) \right)} + 8}{- \frac{63 (4.5850390580)^{2}}{250} - \sin{\left((4.5850390580) \right)}} = 4.5330828135 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%284.5850390580%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B21%20%284.5850390580%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.5850390580%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B63%20%284.5850390580%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%284.5850390580%29%20%5Cright%29%7D%7D%20%3D%204.5330828135%20" alt="LaTeX:  x_{2} =  (4.5850390580) - \frac{- \frac{21 (4.5850390580)^{3}}{250} + \cos{\left((4.5850390580) \right)} + 8}{- \frac{63 (4.5850390580)^{2}}{250} - \sin{\left((4.5850390580) \right)}} = 4.5330828135 " data-equation-content=" x_{2} =  (4.5850390580) - \frac{- \frac{21 (4.5850390580)^{3}}{250} + \cos{\left((4.5850390580) \right)} + 8}{- \frac{63 (4.5850390580)^{2}}{250} - \sin{\left((4.5850390580) \right)}} = 4.5330828135 " /> 
 <img class="equation_image" title=" x_{3} =  (4.5330828135) - \frac{- \frac{21 (4.5330828135)^{3}}{250} + \cos{\left((4.5330828135) \right)} + 8}{- \frac{63 (4.5330828135)^{2}}{250} - \sin{\left((4.5330828135) \right)}} = 4.5323883830 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%284.5330828135%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B21%20%284.5330828135%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.5330828135%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B63%20%284.5330828135%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%284.5330828135%29%20%5Cright%29%7D%7D%20%3D%204.5323883830%20" alt="LaTeX:  x_{3} =  (4.5330828135) - \frac{- \frac{21 (4.5330828135)^{3}}{250} + \cos{\left((4.5330828135) \right)} + 8}{- \frac{63 (4.5330828135)^{2}}{250} - \sin{\left((4.5330828135) \right)}} = 4.5323883830 " data-equation-content=" x_{3} =  (4.5330828135) - \frac{- \frac{21 (4.5330828135)^{3}}{250} + \cos{\left((4.5330828135) \right)} + 8}{- \frac{63 (4.5330828135)^{2}}{250} - \sin{\left((4.5330828135) \right)}} = 4.5323883830 " /> 
 <img class="equation_image" title=" x_{4} =  (4.5323883830) - \frac{- \frac{21 (4.5323883830)^{3}}{250} + \cos{\left((4.5323883830) \right)} + 8}{- \frac{63 (4.5323883830)^{2}}{250} - \sin{\left((4.5323883830) \right)}} = 4.5323882619 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%284.5323883830%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B21%20%284.5323883830%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.5323883830%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B63%20%284.5323883830%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%284.5323883830%29%20%5Cright%29%7D%7D%20%3D%204.5323882619%20" alt="LaTeX:  x_{4} =  (4.5323883830) - \frac{- \frac{21 (4.5323883830)^{3}}{250} + \cos{\left((4.5323883830) \right)} + 8}{- \frac{63 (4.5323883830)^{2}}{250} - \sin{\left((4.5323883830) \right)}} = 4.5323882619 " data-equation-content=" x_{4} =  (4.5323883830) - \frac{- \frac{21 (4.5323883830)^{3}}{250} + \cos{\left((4.5323883830) \right)} + 8}{- \frac{63 (4.5323883830)^{2}}{250} - \sin{\left((4.5323883830) \right)}} = 4.5323882619 " /> 
 <img class="equation_image" title=" x_{5} =  (4.5323882619) - \frac{- \frac{21 (4.5323882619)^{3}}{250} + \cos{\left((4.5323882619) \right)} + 8}{- \frac{63 (4.5323882619)^{2}}{250} - \sin{\left((4.5323882619) \right)}} = 4.5323882619 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%284.5323882619%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B21%20%284.5323882619%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.5323882619%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B63%20%284.5323882619%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%284.5323882619%29%20%5Cright%29%7D%7D%20%3D%204.5323882619%20" alt="LaTeX:  x_{5} =  (4.5323882619) - \frac{- \frac{21 (4.5323882619)^{3}}{250} + \cos{\left((4.5323882619) \right)} + 8}{- \frac{63 (4.5323882619)^{2}}{250} - \sin{\left((4.5323882619) \right)}} = 4.5323882619 " data-equation-content=" x_{5} =  (4.5323882619) - \frac{- \frac{21 (4.5323882619)^{3}}{250} + \cos{\left((4.5323882619) \right)} + 8}{- \frac{63 (4.5323882619)^{2}}{250} - \sin{\left((4.5323882619) \right)}} = 4.5323882619 " /> 
</p> </p>