Math 181 Miniproject 3: Texting Lesson.md
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My lesson Topic
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Hey professor , I am so lost with this derivative stuff. Why do I have to memorize that formula to get it? Can’t I just make a table, plot, and count the squares of rise over run?
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I understand that it can seem a little confusing at first but let’s break it down by each piece. You mentioned rise over run, what does that correlate with? What are you calculating when you use that term?
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I am calculating the slope of a line.
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Great! The slope of a line! Sure, that method is great for when it is a straight line, but what are you going to use when the line has curves and changes frequently?

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I don’t know, I guess that would be hard. I’d have to count for each point.

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Precisely, and lines are infinite that would take you a lifetime and then some. Therefore, we use that formula. First remember, what is the derivative? What does that mean?

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The slope of a line.

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You’re on the right track, the derivative can be interpreted in the following: the slope of the tangent line to F(x) at x= a certain number, or the rate of change at x=a certain number. Let’s say you have a function $F(x)= 2x^2-16x+35$. Does that equation for the line have a clear slope you can distinguish by looking at it, like you would in the formula y=mx+b?

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No, it doesn’t I have no clue what the slope would be.

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This is why that formula is so important. It allows us to be able to get the slope at a specific point and create a formula for the rate of change of the entire function. The formula is called the limit definition of the derivative, the notation for the derivative that we are going to use for now is F’(x) which means F prime of x. You with me so far?

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Yes, so far so good.

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Ok so the formula is $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and let’s say you know have an x value and you input it into the formula. You can calculate the derivative at that certain point, but with each x value that you insert you would get a different number from the limit. This is why we make a new function called the derivative of F using the limit definition.

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So if I use the limit definition of the derivative I can get more then the derivative of a point? I thought we had to input an x value into the formula. What if I don’t know any x values?

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By definition the formula $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ gives us the derivative correct?

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yes

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So, we use that formula and input what we know which is the function. We would solve for (x) so we get an exact formula for F’(x), that way it doesn’t matter what value we use for x we can calculate the derivative with ease!

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Wow! I am mind blown!

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It is very exciting, I agree. The power of the formula truly is limitless 😊

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Lol that was a good one

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Thanks now let us try an example so we can see the definition in action.

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Ok, we’ll see if it really works

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Let’s start with the function of x that I stated above $F(x)= 2x^2-16x+35$ What do you think our first move is?
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To put the function in the formula?

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Excellent! Yes, the first thing we are going to do is plug the function into the definition of the derivative. Remember a key point for the first part of the numerator is that wherever you see an x in the function insert it as x+h.
Look closely on how I entered the function $F(x)= 2x^2-16x+35$ into the definition of the derivative below:

$f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

$f'(x)=\lim_{h \to 0}\frac{2(x+h)^2−16(x+h)+35−(2x^2−16x+35)}{h}$

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Wow that looks really intense.

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Don’t be intimidated, break it down. First expand the $(x+h)^2$. Input that answer into your parenthesis so that you keep your work nice and neat. Then we are going to distribute the numbers in front of the parenthesis. The subtraction sign included!!!

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Ok, give me a second to do the math.

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That’s fine, take your time.

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ok, I'm done

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Great! Check your work with mine below:

$f'(x)=\lim_{h \to 0}\frac{2x^2+4xh+2h−16x−16h+35−2x^2+16x-35}{h}$

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I got that too, I see that a lot of the numbers cancel out

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Fantastic, good inference! So next we are going to cancel out all those numbers. Go ahead and do it and tell me when you’re done.

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Done.

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So, the new formula with everything canceled out looks like this

$f'(x)=\lim_{h \to 0}\frac{4xh+2h^2−16h}{h}$

Is yours the same?

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Yes, it is, can we input h=0 now?

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No, we can’t just plug in h=0, that would give us an error because we cannot divide by zero. Take a close look at the formula we have so far. Do you notice anything about the numerator?

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Oh Yeah! There is an h with every number.

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Yes, but it’s with every term careful to distinguish. This means that we can factor out an h from each term. Go ahead and compute and tell me when you’re done.
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Ok I took the h out.

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So now we our formula is
$f'(x)=\lim_{h \to 0}\frac{h(4x+2h−16)}{h}$
This allows us to cancel the h in the numerator against the h in the denominator.
Giving us
$f'(x)=\lim_{h \to 0}4x+2h−16$

What do you think we do next?

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We can input 0 for h!!!

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Wonderful! Now is when we can input 0 for H, Keep in mind that this final step is when you no longer have to write $f'(x)=\lim_{h \to 0}$ because we are now calculating the derivative.
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Ok, good to know.
So I got 4x−16. Is that right?

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The values are correct but what does that mean?
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Oh haha the whole thing is
$F’(x)= 4x-16$

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So with that formula what can you calculate?
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OMG, I can input any number and get the derivative! You were right!

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Now do you understand why it is so important to know the limit definition of the derivative, you now have access to a plethora of information!
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Yea it’s pretty cool

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I’m happy you understand it now! Some study tips to prepare for these kinds of questions on the exam is practice! You must memorize the formula; with that you have your starting point! Also practice the algebra aspect, being proficient in algebra will help when the terms are a little unpleasant. Lastly, is to properly deal with parenthesis when subtracting in the numerator. Don’t forget to distribute that negative. Mind your work, it’s not a race so take your time.
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Thanks Teach! This helped a lot!

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It was my pleasure if you need additional assistance, feel free to contact me! Enjoy the rest of your day!
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