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{169 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{507 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 3\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (3.0000000000) - \frac{- \frac{169 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{507 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 3.2841927139\end{equation*} \begin{equation*}x_{2} = (3.2841927139) - \frac{- \frac{169 (3.2841927139)^{3}}{1000} + \sin{\left((3.2841927139) \right)} + 6}{- \frac{507 (3.2841927139)^{2}}{1000} + \cos{\left((3.2841927139) \right)}} = 3.2642785341\end{equation*} \begin{equation*}x_{3} = (3.2642785341) - \frac{- \frac{169 (3.2642785341)^{3}}{1000} + \sin{\left((3.2642785341) \right)} + 6}{- \frac{507 (3.2642785341)^{2}}{1000} + \cos{\left((3.2642785341) \right)}} = 3.2641796856\end{equation*} \begin{equation*}x_{4} = (3.2641796856) - \frac{- \frac{169 (3.2641796856)^{3}}{1000} + \sin{\left((3.2641796856) \right)} + 6}{- \frac{507 (3.2641796856)^{2}}{1000} + \cos{\left((3.2641796856) \right)}} = 3.2641796831\end{equation*} \begin{equation*}x_{5} = (3.2641796831) - \frac{- \frac{169 (3.2641796831)^{3}}{1000} + \sin{\left((3.2641796831) \right)} + 6}{- \frac{507 (3.2641796831)^{2}}{1000} + \cos{\left((3.2641796831) \right)}} = 3.2641796831\end{equation*}

Download \(\LaTeX\)

\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{169 x^{3}}{1000} - 6$ using $x_0=3$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{169 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{507 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}  \end{equation*}
Using $x_0 = 3$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} =  (3.0000000000) - \frac{- \frac{169 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{507 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 3.2841927139\end{equation*}
\begin{equation*}x_{2} =  (3.2841927139) - \frac{- \frac{169 (3.2841927139)^{3}}{1000} + \sin{\left((3.2841927139) \right)} + 6}{- \frac{507 (3.2841927139)^{2}}{1000} + \cos{\left((3.2841927139) \right)}} = 3.2642785341\end{equation*}
\begin{equation*}x_{3} =  (3.2642785341) - \frac{- \frac{169 (3.2642785341)^{3}}{1000} + \sin{\left((3.2642785341) \right)} + 6}{- \frac{507 (3.2642785341)^{2}}{1000} + \cos{\left((3.2642785341) \right)}} = 3.2641796856\end{equation*}
\begin{equation*}x_{4} =  (3.2641796856) - \frac{- \frac{169 (3.2641796856)^{3}}{1000} + \sin{\left((3.2641796856) \right)} + 6}{- \frac{507 (3.2641796856)^{2}}{1000} + \cos{\left((3.2641796856) \right)}} = 3.2641796831\end{equation*}
\begin{equation*}x_{5} =  (3.2641796831) - \frac{- \frac{169 (3.2641796831)^{3}}{1000} + \sin{\left((3.2641796831) \right)} + 6}{- \frac{507 (3.2641796831)^{2}}{1000} + \cos{\left((3.2641796831) \right)}} = 3.2641796831\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 \sin{\left(x \right)}= \frac{169 x^{3}}{1000} - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B169%20x%5E%7B3%7D%7D%7B1000%7D%20-%206%20" alt="LaTeX:  \displaystyle \sin{\left(x \right)}= \frac{169 x^{3}}{1000} - 6 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{169 x^{3}}{1000} - 6 " />  using  <img class="equation_image" title=" \displaystyle x_0=3 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D3%20" alt="LaTeX:  \displaystyle x_0=3 " data-equation-content=" \displaystyle x_0=3 " /> . </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{169 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{507 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B169%20x_%7Bn%7D%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B507%20x_%7Bn%7D%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%7D%20%20%20" alt="LaTeX:  x_{n+1} =  x_{n} - \frac{- \frac{169 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{507 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{169 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{507 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}   " /> 
Using  <img class="equation_image" title=" \displaystyle x_0 = 3 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%203%20" alt="LaTeX:  \displaystyle x_0 = 3 " data-equation-content=" \displaystyle x_0 = 3 " />  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} =  (3.0000000000) - \frac{- \frac{169 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{507 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 3.2841927139 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%283.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B169%20%283.0000000000%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B507%20%283.0000000000%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%7D%20%3D%203.2841927139%20" alt="LaTeX:  x_{1} =  (3.0000000000) - \frac{- \frac{169 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{507 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 3.2841927139 " data-equation-content=" x_{1} =  (3.0000000000) - \frac{- \frac{169 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{507 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 3.2841927139 " /> 
 <img class="equation_image" title=" x_{2} =  (3.2841927139) - \frac{- \frac{169 (3.2841927139)^{3}}{1000} + \sin{\left((3.2841927139) \right)} + 6}{- \frac{507 (3.2841927139)^{2}}{1000} + \cos{\left((3.2841927139) \right)}} = 3.2642785341 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%283.2841927139%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B169%20%283.2841927139%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.2841927139%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B507%20%283.2841927139%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.2841927139%29%20%5Cright%29%7D%7D%20%3D%203.2642785341%20" alt="LaTeX:  x_{2} =  (3.2841927139) - \frac{- \frac{169 (3.2841927139)^{3}}{1000} + \sin{\left((3.2841927139) \right)} + 6}{- \frac{507 (3.2841927139)^{2}}{1000} + \cos{\left((3.2841927139) \right)}} = 3.2642785341 " data-equation-content=" x_{2} =  (3.2841927139) - \frac{- \frac{169 (3.2841927139)^{3}}{1000} + \sin{\left((3.2841927139) \right)} + 6}{- \frac{507 (3.2841927139)^{2}}{1000} + \cos{\left((3.2841927139) \right)}} = 3.2642785341 " /> 
 <img class="equation_image" title=" x_{3} =  (3.2642785341) - \frac{- \frac{169 (3.2642785341)^{3}}{1000} + \sin{\left((3.2642785341) \right)} + 6}{- \frac{507 (3.2642785341)^{2}}{1000} + \cos{\left((3.2642785341) \right)}} = 3.2641796856 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%283.2642785341%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B169%20%283.2642785341%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.2642785341%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B507%20%283.2642785341%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.2642785341%29%20%5Cright%29%7D%7D%20%3D%203.2641796856%20" alt="LaTeX:  x_{3} =  (3.2642785341) - \frac{- \frac{169 (3.2642785341)^{3}}{1000} + \sin{\left((3.2642785341) \right)} + 6}{- \frac{507 (3.2642785341)^{2}}{1000} + \cos{\left((3.2642785341) \right)}} = 3.2641796856 " data-equation-content=" x_{3} =  (3.2642785341) - \frac{- \frac{169 (3.2642785341)^{3}}{1000} + \sin{\left((3.2642785341) \right)} + 6}{- \frac{507 (3.2642785341)^{2}}{1000} + \cos{\left((3.2642785341) \right)}} = 3.2641796856 " /> 
 <img class="equation_image" title=" x_{4} =  (3.2641796856) - \frac{- \frac{169 (3.2641796856)^{3}}{1000} + \sin{\left((3.2641796856) \right)} + 6}{- \frac{507 (3.2641796856)^{2}}{1000} + \cos{\left((3.2641796856) \right)}} = 3.2641796831 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%283.2641796856%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B169%20%283.2641796856%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.2641796856%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B507%20%283.2641796856%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.2641796856%29%20%5Cright%29%7D%7D%20%3D%203.2641796831%20" alt="LaTeX:  x_{4} =  (3.2641796856) - \frac{- \frac{169 (3.2641796856)^{3}}{1000} + \sin{\left((3.2641796856) \right)} + 6}{- \frac{507 (3.2641796856)^{2}}{1000} + \cos{\left((3.2641796856) \right)}} = 3.2641796831 " data-equation-content=" x_{4} =  (3.2641796856) - \frac{- \frac{169 (3.2641796856)^{3}}{1000} + \sin{\left((3.2641796856) \right)} + 6}{- \frac{507 (3.2641796856)^{2}}{1000} + \cos{\left((3.2641796856) \right)}} = 3.2641796831 " /> 
 <img class="equation_image" title=" x_{5} =  (3.2641796831) - \frac{- \frac{169 (3.2641796831)^{3}}{1000} + \sin{\left((3.2641796831) \right)} + 6}{- \frac{507 (3.2641796831)^{2}}{1000} + \cos{\left((3.2641796831) \right)}} = 3.2641796831 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%283.2641796831%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B169%20%283.2641796831%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.2641796831%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B507%20%283.2641796831%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.2641796831%29%20%5Cright%29%7D%7D%20%3D%203.2641796831%20" alt="LaTeX:  x_{5} =  (3.2641796831) - \frac{- \frac{169 (3.2641796831)^{3}}{1000} + \sin{\left((3.2641796831) \right)} + 6}{- \frac{507 (3.2641796831)^{2}}{1000} + \cos{\left((3.2641796831) \right)}} = 3.2641796831 " data-equation-content=" x_{5} =  (3.2641796831) - \frac{- \frac{169 (3.2641796831)^{3}}{1000} + \sin{\left((3.2641796831) \right)} + 6}{- \frac{507 (3.2641796831)^{2}}{1000} + \cos{\left((3.2641796831) \right)}} = 3.2641796831 " /> 
</p> </p>