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Math 181 Miniproject 3: Texting Lesson.md
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My lesson Topic
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<div><img class="left"/><div class="alert gray">

Hey happy emoji! I had a question for you. Can you explain to me how to find local linearization? I do not quite understand it. 

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

Of course panic emoji! First you have to understand that local linearization just means to find an equation by using the tangent line equation.

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


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

I see, but how do we go from the tangent line equation to the local linearzation equation?

</div></div> 

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

I really do not know how we get from $y-y_{o}=m\left(x-x_{o}\right)$ to $L\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)$

</div></div> 

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

These are the steps to get to local linearization:
$y-y_{o}=m\left(x-x_{o}\right)$
$y=y_{o}+m\left(x-x_{o}\right)$
Then you change the variables to the function, and the derivitive of the function. So:
$L\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)$
$a=$ the point of the function

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

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

Okay! Great! I am starting to understand. can you give me an example problem?

</div></div> 

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

Sure thing. Find the linear approximation to $h\left(t\right)=t^{4}-6t^{3}+3t-7$ at $t=-3$ 
First we can find $h(t)$ by using the function. So $h(-3)=227$
Then we find $h'(-3)$ by using the derivitive definition, which you should already know so I will cut to the chase. $h'(-3)=-267$. Now we can plug in what we know to the formula: $L\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)$
So the formula will now look like this:
$h\left(t\right)=h\left(-3\right)+h'\left(-3\right)\left(x-(-3)\right)$ 
And then:
$h\left(t\right)=227-267\left(x+3\right)$
This will be our formula for linear approximation. Leave it as is, do not simplify. 



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

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

Okay! I think I get it. Any more tips before I take this test?

</div></div> 

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

Yes! Please remember that the equation you find is only good for points close to the number you found it for. For example the equation I showed you: $h\left(t\right)=227-267\left(x+3\right)$ 
Would only be good for numbers close to -3, like -3.1 and so on. If you input a number far from -3, like 14 then you are going to get a pretty hefty percent error. 
I hope my tips helped you out! Good luck on your test!

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


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