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Find the difference quotient of \(\displaystyle f(x)=7 x^{2} + 3 x - 2\) .

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

Download \(\LaTeX\) 

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

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the difference quotient of  <img class="equation_image" title=" \displaystyle f(x)=7 x^{2} + 3 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D7%20x%5E%7B2%7D%20%2B%203%20x%20-%202%20" alt="LaTeX:  \displaystyle f(x)=7 x^{2} + 3 x - 2 " data-equation-content=" \displaystyle f(x)=7 x^{2} + 3 x - 2 " />  . </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)=3 h + 3 x + 7 \left(h + x\right)^{2} - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D3%20h%20%2B%203%20x%20%2B%207%20%5Cleft%28h%20%2B%20x%5Cright%29%5E%7B2%7D%20-%202%20" alt="LaTeX:  \displaystyle f(x+h)=3 h + 3 x + 7 \left(h + x\right)^{2} - 2 " data-equation-content=" \displaystyle f(x+h)=3 h + 3 x + 7 \left(h + x\right)^{2} - 2 " />  and expanding gives  <img class="equation_image" title=" \displaystyle f(x+h)=7 h^{2} + 14 h x + 3 h + 7 x^{2} + 3 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D7%20h%5E%7B2%7D%20%2B%2014%20h%20x%20%2B%203%20h%20%2B%207%20x%5E%7B2%7D%20%2B%203%20x%20-%202%20" alt="LaTeX:  \displaystyle f(x+h)=7 h^{2} + 14 h x + 3 h + 7 x^{2} + 3 x - 2 " data-equation-content=" \displaystyle f(x+h)=7 h^{2} + 14 h x + 3 h + 7 x^{2} + 3 x - 2 " />   Evaluating the difference quotient gives  <img class="equation_image" title=" \displaystyle \frac{(7 h^{2} + 14 h x + 3 h + 7 x^{2} + 3 x - 2)-(7 x^{2} + 3 x - 2)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%287%20h%5E%7B2%7D%20%2B%2014%20h%20x%20%2B%203%20h%20%2B%207%20x%5E%7B2%7D%20%2B%203%20x%20-%202%29-%287%20x%5E%7B2%7D%20%2B%203%20x%20-%202%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{(7 h^{2} + 14 h x + 3 h + 7 x^{2} + 3 x - 2)-(7 x^{2} + 3 x - 2)}{h} " data-equation-content=" \displaystyle \frac{(7 h^{2} + 14 h x + 3 h + 7 x^{2} + 3 x - 2)-(7 x^{2} + 3 x - 2)}{h} " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \frac{7 h^{2} + 14 h x + 3 h}{h}=7 h + 14 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B7%20h%5E%7B2%7D%20%2B%2014%20h%20x%20%2B%203%20h%7D%7Bh%7D%3D7%20h%20%2B%2014%20x%20%2B%203%20" alt="LaTeX:  \displaystyle \frac{7 h^{2} + 14 h x + 3 h}{h}=7 h + 14 x + 3 " data-equation-content=" \displaystyle \frac{7 h^{2} + 14 h x + 3 h}{h}=7 h + 14 x + 3 " />  </p> </p>