\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use the table below to estimate the velocity at \(\displaystyle t=8\) by averaging the slopes of the two adjacent secant lines corresponding to the two points closest to \(\displaystyle (8,22)\)
Using the point to the left gives \(\displaystyle m_1 = \frac{22-15}{8-4} = \frac{7}{4}\). Using the point to the right gives \(\displaystyle m_2 = \frac{22-31}{8-12} = \frac{9}{4}\). Taking the average of the slopes gives \(\displaystyle \frac{\frac{7}{4}+\frac{9}{4}}{2} = 2\). The estimate of the velocity is \(\displaystyle 2\) meters per second.
\begin{question}Use the table below to estimate the velocity at $t=8$ by averaging the slopes of the two adjacent secant lines corresponding to the two points closest to $(8,22)$\newline \begin{tabular}{|c|c|c|c|c|c|}\hline $t$ seconds & 0 & 4 & 8 & 12 & 16 \\ \hline $x$ meters & 10 & 15 & 22 & 31 & 39 \\ \hline \end{tabular}\newline \soln{9cm}{Using the point to the left gives $m_1 = \frac{22-15}{8-4} = \frac{7}{4}$. Using the point to the right gives $m_2 = \frac{22-31}{8-12} = \frac{9}{4}$. Taking the average of the slopes gives $\frac{\frac{7}{4}+\frac{9}{4}}{2} = 2$. The estimate of the velocity is $2$ meters per second.} \end{question}
\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}
<p> <p>Use the table below to estimate the velocity at <img class="equation_image" title=" \displaystyle t=8 " src="/equation_images/%20%5Cdisplaystyle%20t%3D8%20" alt="LaTeX: \displaystyle t=8 " data-equation-content=" \displaystyle t=8 " /> by averaging the slopes of the two adjacent secant lines corresponding to the two points closest to <img class="equation_image" title=" \displaystyle (8,22) " src="/equation_images/%20%5Cdisplaystyle%20%288%2C22%29%20" alt="LaTeX: \displaystyle (8,22) " data-equation-content=" \displaystyle (8,22) " /> <br>
</p> </p>
<p> <p>Using the point to the left gives <img class="equation_image" title=" \displaystyle m_1 = \frac{22-15}{8-4} = \frac{7}{4} " src="/equation_images/%20%5Cdisplaystyle%20m_1%20%3D%20%5Cfrac%7B22-15%7D%7B8-4%7D%20%3D%20%5Cfrac%7B7%7D%7B4%7D%20" alt="LaTeX: \displaystyle m_1 = \frac{22-15}{8-4} = \frac{7}{4} " data-equation-content=" \displaystyle m_1 = \frac{22-15}{8-4} = \frac{7}{4} " /> . Using the point to the right gives <img class="equation_image" title=" \displaystyle m_2 = \frac{22-31}{8-12} = \frac{9}{4} " src="/equation_images/%20%5Cdisplaystyle%20m_2%20%3D%20%5Cfrac%7B22-31%7D%7B8-12%7D%20%3D%20%5Cfrac%7B9%7D%7B4%7D%20" alt="LaTeX: \displaystyle m_2 = \frac{22-31}{8-12} = \frac{9}{4} " data-equation-content=" \displaystyle m_2 = \frac{22-31}{8-12} = \frac{9}{4} " /> . Taking the average of the slopes gives <img class="equation_image" title=" \displaystyle \frac{\frac{7}{4}+\frac{9}{4}}{2} = 2 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%5Cfrac%7B7%7D%7B4%7D%2B%5Cfrac%7B9%7D%7B4%7D%7D%7B2%7D%20%3D%202%20" alt="LaTeX: \displaystyle \frac{\frac{7}{4}+\frac{9}{4}}{2} = 2 " data-equation-content=" \displaystyle \frac{\frac{7}{4}+\frac{9}{4}}{2} = 2 " /> . The estimate of the velocity is <img class="equation_image" title=" \displaystyle 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX: \displaystyle 2 " data-equation-content=" \displaystyle 2 " /> meters per second.</p> </p>