Math 181 Miniproject 3: Texting Lesson.md
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My lesson Topic
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<style>
body {
background-color: #eeeeee;
}

h1 {
color: pink;
margin-left: 40px;
}

.gray {
margin-left: 50px ;
margin-right: 29%;
font-weight: 500;
color: #000000;
background-color: #cccccc;
border-color: #aaaaaa;
}

.blue {
display: inline-block;
margin-left: 29% ;
margin-right: 0%;
width: -webkit-calc(70% - 50px);
width: -moz-calc(70% - 50px);
width: calc(70% - 50px);
font-weight: 500;
color: #fff;
border-color: #336699;
background-color: #337799;
}

.left {
content:url("https://i.imgur.com/rUsxo7j.png");
width:50px;
border-radius: 50%;
float:left;
}
.right{
content:url("https://i.imgur.com/5ALcyl3.png"); width:50px;
border-radius: 50%;
display: inline-block;
vertical-align:top;
}
</style>

<div id="container" style=" padding: 6px;
color: #fff;
border-color: #336699;
background-color: #337799;
display: flex;
justify-content: space-between;
margin-bottom:3px;">
<div>
<i class="fa fa-envelope fa-2x"></i>
</div>
<div>
<i class="fa fa-camera fa-2x"></i>
</div>

<div>
<i class="fa fa-comments fa-2x"></i>
</div>
<div>
<i class="fa fa-address-card fa-2x" aria-hidden="true"></i>
</div>
<div>
<i class="fa fa-phone fa-2x" aria-hidden="true"></i>
</div>
<div>
<i class="fa fa-list-ul fa-2x" aria-hidden="true"></i>
</div>
<div>
<i class="fa fa-user-plus fa-2x" aria-hidden="true"></i>
</div>
</div>

<div><img class="left"/><div class="alert gray">

I was doing my homework last night and I got stuck on this problem, can you help me through it? The problem is "For the function $f(x)=4x-x^2+22$ find the exact formula for f'(x). Use only the defintion of the derivative."

</div></div>

<div><div class="alert blue">

Yes, I'm glad to help you through this problem. The first step to solve this problem is to realize that you will be using the formula $\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

I see. So I just need to plug in the formula that was given in the question in order to find f'(x).

</div></div>

<div><img class="left"/><div class="alert gray">

Like this,
$f(x)=4x-x^2+22$
$f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$
$f'(x)=\lim_{h \to 0}\frac{(4(x+h)-(x+h)^2+22)-(4x-x^2+22)}{h}$
$f'(x)=\lim_{h \to 0}\frac{4x+4h-x^2-2xh-h^2+22-4x+x^2-22}{h}$
$f'(x)=\lim_{h \to 0}\frac{4h-2xh-h^2}{h}$
$f'(x)=\lim_{h \to 0}\frac{h(4-2x-h)}{h}$
$f'(x)=\lim_{h \to 0}4-2x-h$
$f'(x)=\lim_{h \to 0}4-2x-0$
$f'(x)=4-2x$

</div></div>

<div><div class="alert blue">

Yes you are correct, and this type of problem is important because it is necessary to know how to find the derivative or slope of any given function to succeed in calculus.

</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

Thank you for helping me solve this problem, do you have any helpful tips for when I take the exam.

</div></div>

<div><div class="alert blue">

A helpful tip for when you are solving the derivative of a function using the limit definition is to make sure you keep your work clear as possible, that way on the exam the professor will be able to understand your process and if you make any small errors, points may be saved due to clarity.

</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

Thank you for helping me. :)

</div></div>

<div><div class="alert blue">

No problem! I'm here to help.

</div><img class="right"/></div>

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