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: skyblue;
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">

So... what exactly is this assignment? it looks tricky.

</div></div>

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

The question is to consider the function,
$f\left(x\right)=5x^{2}+2x+10$
and using only the limit definition of the derivative, find the exact formula for f'(x)

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

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

Okay, so how do I start?

</div></div>

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

To start, you need to know the formula for the limit definition of the derivative f'(x) of a function f(x)

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

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

I see, and what is that formula?

</div></div>

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

The formula of the limit definition of a derivative is
$$f'\left(x\right)=\lim_{h→0}\ \frac{\left[f\left(x+h\right)-f\left(x\right)\right]}{h}$$

this is something that you just need to memorize.

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

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

Alright, what next?

</div></div>

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

Once you have the formula down, you can plug in the function we were given into the limit definition function
$$f'\left(x\right)=\lim_{h→0}\ \frac{\left[\left[5\left(x+h\right)^{2}+2\left(x+h\right)+10\right]-\left[5x^{2}+2x+10\right]\right]}{h}$$

After plugging in the function, factor and simplify the numerator and cancel out like terms if possible

$$f'\left(x\right)=\lim_{h→0}\frac{\left[5x^{2}+10xh+5h^{2}+2x+2h+10-5x^{2}-2x-10\right]}{h}$$

in this numerator, the $5x^2$, 2x, and 10 cancel out leaving the formula

$$f'\left(x\right)=\lim_{h→0}\ \frac{\left(10xh+5h^{2}+2h\right)}{h}$$

then you can factor out and h from the numerator to simplify further

$$f'\left(x\right)=\lim_{h→0}\ \frac{h\left(10x+5h+2\right)}{h}$$
then you can cancel out the h from the front of the numerator and the h in the denominator, this will leave the final function

$$f'\left(x\right)=\lim_{h→0}\ 10x+5h+2$$

once you have the final simplified function, you can plug in 0 for all of the h values since you are finding the limit as h approaches 0. This gives you:
$$10x+5\left(0\right)+2$$ or $$10x+2$$

so, the answer for the formula of f'(x) is $$f'(x)=10x+2$$
using this formula you can plug in any value for x to find the limit of the function $f(x)=5x^2+2x+10$
</div><img class="right"/></div>

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

Oh, I see now. Thank you!

</div></div>

</div></div>

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

You're welcome!

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