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The difference quotient is \(\displaystyle \frac{f(x+h)-f(x)}{h}\). Evaluating \(\displaystyle f(x+h)=- 7 h - 7 x - 7 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} + 1\) and expanding gives \(\displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1\) Evaluating the difference quotient gives \(\displaystyle \frac{(- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1)-(- 7 x^{3} + 7 x^{2} - 7 x + 1)}{h}\) Simplifying gives \(\displaystyle \frac{- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h}{h}=- 7 h^{2} - 21 h x + 7 h - 21 x^{2} + 14 x - 7\)

Download \(\LaTeX\)

\begin{question}Find the difference quotient of $f(x)=- 7 x^{3} + 7 x^{2} - 7 x + 1$ . 
    \soln{9cm}{The difference quotient is $\frac{f(x+h)-f(x)}{h}$. Evaluating $f(x+h)=- 7 h - 7 x - 7 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} + 1$ and expanding gives $f(x+h)=- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1$  Evaluating the difference quotient gives $\frac{(- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1)-(- 7 x^{3} + 7 x^{2} - 7 x + 1)}{h}$ Simplifying gives $\frac{- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h}{h}=- 7 h^{2} - 21 h x + 7 h - 21 x^{2} + 14 x - 7$ }

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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

\end{question}\end{document}

HTML for Canvas

<p> <p>Find the difference quotient of  <img class="equation_image" title=" \displaystyle f(x)=- 7 x^{3} + 7 x^{2} - 7 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%207%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20-%207%20x%20%2B%201%20" alt="LaTeX:  \displaystyle f(x)=- 7 x^{3} + 7 x^{2} - 7 x + 1 " data-equation-content=" \displaystyle f(x)=- 7 x^{3} + 7 x^{2} - 7 x + 1 " />  . </p> </p>

HTML for Canvas

<p> <p>The difference quotient is  <img class="equation_image" title=" \displaystyle \frac{f(x+h)-f(x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{f(x+h)-f(x)}{h} " data-equation-content=" \displaystyle \frac{f(x+h)-f(x)}{h} " /> . Evaluating  <img class="equation_image" title=" \displaystyle f(x+h)=- 7 h - 7 x - 7 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%207%20h%20-%207%20x%20-%207%20%5Cleft%28h%20%2B%20x%5Cright%29%5E%7B3%7D%20%2B%207%20%5Cleft%28h%20%2B%20x%5Cright%29%5E%7B2%7D%20%2B%201%20" alt="LaTeX:  \displaystyle f(x+h)=- 7 h - 7 x - 7 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} + 1 " data-equation-content=" \displaystyle f(x+h)=- 7 h - 7 x - 7 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} + 1 " />  and expanding gives  <img class="equation_image" title=" \displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%207%20h%5E%7B3%7D%20-%2021%20h%5E%7B2%7D%20x%20%2B%207%20h%5E%7B2%7D%20-%2021%20h%20x%5E%7B2%7D%20%2B%2014%20h%20x%20-%207%20h%20-%207%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20-%207%20x%20%2B%201%20" alt="LaTeX:  \displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1 " data-equation-content=" \displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1 " />   Evaluating the difference quotient gives  <img class="equation_image" title=" \displaystyle \frac{(- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1)-(- 7 x^{3} + 7 x^{2} - 7 x + 1)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%28-%207%20h%5E%7B3%7D%20-%2021%20h%5E%7B2%7D%20x%20%2B%207%20h%5E%7B2%7D%20-%2021%20h%20x%5E%7B2%7D%20%2B%2014%20h%20x%20-%207%20h%20-%207%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20-%207%20x%20%2B%201%29-%28-%207%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20-%207%20x%20%2B%201%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{(- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1)-(- 7 x^{3} + 7 x^{2} - 7 x + 1)}{h} " data-equation-content=" \displaystyle \frac{(- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h - 7 x^{3} + 7 x^{2} - 7 x + 1)-(- 7 x^{3} + 7 x^{2} - 7 x + 1)}{h} " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \frac{- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h}{h}=- 7 h^{2} - 21 h x + 7 h - 21 x^{2} + 14 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B-%207%20h%5E%7B3%7D%20-%2021%20h%5E%7B2%7D%20x%20%2B%207%20h%5E%7B2%7D%20-%2021%20h%20x%5E%7B2%7D%20%2B%2014%20h%20x%20-%207%20h%7D%7Bh%7D%3D-%207%20h%5E%7B2%7D%20-%2021%20h%20x%20%2B%207%20h%20-%2021%20x%5E%7B2%7D%20%2B%2014%20x%20-%207%20" alt="LaTeX:  \displaystyle \frac{- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h}{h}=- 7 h^{2} - 21 h x + 7 h - 21 x^{2} + 14 x - 7 " data-equation-content=" \displaystyle \frac{- 7 h^{3} - 21 h^{2} x + 7 h^{2} - 21 h x^{2} + 14 h x - 7 h}{h}=- 7 h^{2} - 21 h x + 7 h - 21 x^{2} + 14 x - 7 " />  </p> </p>