WEBVTT



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Okay, a little review from

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geometry.

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What do we call a line that goes through

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a circle and touches two different points?

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What's that?

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Secant line.

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And what do

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we call a

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line that

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touches it at one single point?

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tangent line. Let's call it tangent line.

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Why are these also the name of two of our

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trig functions?

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Is that just random or...

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It does have something to do with secant and tangent

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lines but we'll get to that some other day.

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Um...

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What's a word close to tangent that means

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something similar?

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Sorry, not a mathematical word.

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An English word.

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tangent line.

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tangent line.

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Maybe the circle looks something like a tangent line.

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No, I'm looking for something else.

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I look for the word tangible.

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What does tangible mean?

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You can touch it.

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So it just touches it one time.

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So I think that's where the word comes from.

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Well the same is true for functions.

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The

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If a line crosses through a function at two different

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points, it's called a secant line.

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And if line touches

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at one single point,

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okay,

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it's called the tangent line.

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But it's a little bit harder to define.

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For a circle, it's very simple.

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A line that touches a circle just once is the

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tangent line. You can't say the same here because this

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line right here touches the function of one single point.

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It's not tangent.

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Okay, so it's a little harder to define what it

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means to be tangent.

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Okay, basically it has to be going in the exact

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same direction at

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that point.

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I have a very similar slope to some line the

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secant line goes through two points

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We'll get to the actual definition of it.

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Thank you. Bye.

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Most of you have a basic understanding

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What if we want to find the slope of the

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secant line

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What will we need?

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The two ordered pairs, right?

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Let's call this x1,

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xy1. This

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x2, y2. And

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then the slope

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is what for me?

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y2 minus y1 over x2 minus x1.

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Okay,

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write the rise over the run.

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What about the slope of the secant line?

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Sorry, slope of the tangent line.

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all we have is one single point

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let's call it x1 y1

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is there a nice formula for it?

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all we have is one point

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and we don't know the slope

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that's if we want to find the equation of that

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line yeah how do we even find the slope

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is it just y1 over x1

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definitely not

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definitely not

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Okay, so who knows?

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Okay, we're going to have to do some work and

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eventually we're going to figure out how to do it.

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But this is the major part of calculus,

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uh, is how in the world are we going to

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do this?

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Slopes of lines are often called rates,

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rate of change.

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What is the definition of a rate?

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Anybody remember from seventh grade?

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What's the definition of a rate?

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No thoughts?

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Just the word rate.

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For example,

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uh... 50 miles per

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hour is a rate.

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Okay.

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It is a ratio, but there's one more

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thing.

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So it's a ratio

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of

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two quantities.

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units. Very important with different

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units.

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For example, like two-thirds is a ratio but it's

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not a rate. Why is it not a rate?

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There are no units or they were the same units

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but two cups per

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three

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Gallons

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That's not really good here, let's go three cups

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of flour

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two, three gallons

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of water or something like

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that. So as long as you have different units.

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50 miles per hour, same thing as 50 miles

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per one hour. You gotta have two

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different units.

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That's a rate.

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Okay, going back here, the

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slope here of a line is a rate.

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Let's just say the slope of this line is

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2

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thirds.

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Just making up some number.

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I don't see units.

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Why is it still a rate?

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Because it's such a new y on the checkbook.

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Yeah, x has some units.

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We're studying writing down.

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and y have some units I sit right down so

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it's two y units

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to three x

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units

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okay sometimes we're given a the units sometimes we're

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not but it is always a rate okay and we

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sometimes call a rate of change because it's how fast

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the X is changing compared to how fast the y

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is changing. Okay,

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what are some examples of rates in real life?

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I gave you one already, 50 miles per hour.

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Discuss with somebody next to you, What are examples?

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Giving an actual number and then the units.

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Yeah, row one. Five cookies for ten dollars.

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Okay!

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Five...

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cookies... to ten dollars.

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Okay, row two, somebody give me one.

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Or in other words, one tablespoon

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per gallon.

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Same thing. Okay. Is that how much you pee for

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fish? I don't have fish.

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You don't have fish?

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Do you have fish?

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No.

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That's cool. Interesting. Somebody wrote three.

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$20 an hour. Ooh, okay.

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And what rate would

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that be?

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What's that? Your wage, often known as your

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hourly rate.

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Five ice cream scoop a minute.

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That's the rate at which you can scoop ice cream.

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Okay. I like that.

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Do you eat a lot of ice cream?

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and what would be that rate of?

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population density

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yeah I like that row 6

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100 words per minute ooh I like that

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Okay, and what rate is that the rate of?

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Like, typing. I see you're typing speed.

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Speed is always a type of rate.

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Rate of change, same thing.

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That's pretty fast.

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Okay, anybody else? Give me some more examples.

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throw something

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out.

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100

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frames per

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second and what's what's

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that the rate of anybody?

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So that'd be a frame rate how fast your TV

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screen or a camera refreshes

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or takes pictures so many frames per every

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second.

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Somebody else something.

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$100,000 a year.

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It's your yearly rate of pay.

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Also your salary maybe.

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You make a lot of money.

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It's pretty good what else

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we got

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and what rate would that be interest rates

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your interest rate okay depending How many years and how

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many dollars you get a certain?

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percentage that per year

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like that anybody else This the type

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of speed

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These are all different speeds, but not about distance

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But

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We have

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velocity and speed versus speed

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uhh we all taken some of you have had physics

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already?

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so what's the difference between velocity and speed?

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what's the term they usually use in physics?

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well they typically use the word velocity

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so what is the difference between velocity and speed?

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The speed is velocity with no

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direction.

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So velocity is speed and

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direction.

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Okay.

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And they typically deal with vectors and direction.

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For us, in this class,

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you only have two directions.

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Either the positive direction...

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...or the negative direction.

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Okay.

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Everything we do with velocity in this class, Whatever

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is moving only moves on a single line.

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It's either moving in the positive direction or it's moving

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in the negative direction.

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Everything in its class has to do with velocity and

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speed. It's always on a straight line.

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OK, this is very important.

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And we either... So if you are negative

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five miles per hour...

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Okay, that means you're going backwards or in the negative

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direction at 5 mph.

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Okay, so speed is very simple.

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it is the

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absolute value of velocity.

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Okay, this is very important.

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So if your velocity is negative five miles per hour

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What's your speed?

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five miles per

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Okay, if you're

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32 feet per second

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Feet per second

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This is if this is your velocity, what's your speed?

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32 feet direction. Okay, if it's positive you simply are

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going to the right on a number line

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everything here we move on a number line okay it

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could be on a vertical number line up and down

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horizontal number line just everything's on a number line and

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velocity can be negative just being you're going backwards

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Okay.

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a position

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function.

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OK. So anytime you're given a function,

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typically an equation,

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and if the y variable is some distance

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and the x variable is measured in any kind of

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time period, it is a position function.

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Okay.

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How do I know this is a position function?

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The x is in some time, measured in minutes, and

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the y is some distance, in this case measured in

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years.

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Let's say this is me walking or something.

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So this is Mr.

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Whitman's position

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at

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any time

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t in minutes.

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Okay, what we don't see is some the line that

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I'm walking on.

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It's some other line that has nothing to do with

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this graph, okay?

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Okay at time zero where am I?

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What's that?

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and time, time, time, 10 meters.

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So at time 0, which is the x-axis or

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t-axis,

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I'm exactly here. So I am right here

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at time 0.

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And at time,

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oh, yeah.

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These are all in meters.

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And at time 10 minutes, where am I at?

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I'm right here

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Okay

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Okay, and then I was during that ten minutes I

00:18:34.620 --> 00:18:35.480
was moving

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What was my average

00:18:42.500 --> 00:18:43.300
Velocity

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How do

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we...

00:18:58.620 --> 00:19:02.240
So average velocity is change total

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change

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in distance

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divided by

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my total change in

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How did you get that?

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I did it at 70 minus 10 because it started

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at 10. And then 10 minus 0.

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So that's 60 per...

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70 minus 10 meters.

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And 10 minus 0

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minutes.

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is 60 over 10

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which is 6 and the units are

00:19:54.920 --> 00:19:59.160
meters per minute

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that's pretty slow walker by the

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way.

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Okay, which is exactly

00:20:19.860 --> 00:20:21.600
the slope of this line.

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Okay, your average velocity is

00:20:27.840 --> 00:20:31.660
the same thing as the slope of the secant line.

00:20:33.360 --> 00:20:36.400
Was I always the whole time

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moving at 6 meters per minute?

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How do we know?

00:20:46.780 --> 00:20:49.080
The lines not linear

00:20:49.980 --> 00:20:54.420
Okay, if this were my position function right here this

00:20:54.420 --> 00:20:58.880
red line Then it would always be moving exactly six

00:20:58.880 --> 00:21:00.120
meters per second

00:21:02.620 --> 00:21:05.620
Let's say at one minute mark

00:21:06.720 --> 00:21:11.160
oops Was it moving faster than six or slower

00:21:11.160 --> 00:21:11.810
than six?

00:21:14.240 --> 00:21:16.260
Faster and how can we tell?

00:21:17.960 --> 00:21:19.280
Well if

00:21:22.100 --> 00:21:25.560
If we found that slope with a secant line between

00:21:25.560 --> 00:21:28.880
zero and one minute, it's way steeper, right?

00:21:29.400 --> 00:21:32.140
So over that whole minute I was going way faster

00:21:34.060 --> 00:21:36.560
But what about exactly at one minute

00:21:43.500 --> 00:21:45.780
Well, what we could do is

00:21:46.820 --> 00:21:50.700
Find the slope of two points real close to each

00:21:50.700 --> 00:21:51.350
other

00:21:53.440 --> 00:21:55.040
And let's say it was that.

00:21:59.120 --> 00:22:01.160
You can keep getting closer and closer and closer.

00:22:02.360 --> 00:22:05.940
And it's going to be a steeper slope than this

00:22:05.940 --> 00:22:08.460
slope. So at one minute I was going faster.

00:22:10.280 --> 00:22:14.340
Ideally we have to find the slope of the

00:22:14.340 --> 00:22:15.360
tangent line.

00:22:16.420 --> 00:22:20.260
The slope of the tangent line is your instantaneous

00:22:20.260 --> 00:22:24.860
velocity. Okay, so for

00:22:24.860 --> 00:22:28.240
any position function, the instantaneous

00:22:28.240 --> 00:22:33.160
velocity

00:22:33.160 --> 00:22:35.300
is the

00:22:35.300 --> 00:22:39.900
slope of the

00:22:39.900 --> 00:22:40.880
tangent line.

00:22:42.580 --> 00:22:46.600
What we can't do is, well, I'll show you in

00:22:46.600 --> 00:22:48.400
a second. I started right here.

00:22:49.200 --> 00:22:52.740
At some point I was moving faster and then I

00:22:52.740 --> 00:22:54.380
slowed down and almost stopped.

00:22:55.240 --> 00:22:59.520
Because like around here, the slope of the tangent line

00:22:59.520 --> 00:23:02.220
is practically zero.

00:23:03.740 --> 00:23:05.500
So I, wherever I was at.

00:23:08.620 --> 00:23:12.500
Nice little way down, I went real slow, real slow,

00:23:12.660 --> 00:23:14.860
and then I sped up at the end.

00:23:15.840 --> 00:23:18.160
Okay, it's hard to do it moving.

00:23:18.160 --> 00:23:22.580
Let's uh... Okay, here's an example of a

00:23:22.580 --> 00:23:26.360
position function. The function, I think it's x times sine

00:23:26.360 --> 00:23:27.240
of x.

00:23:28.660 --> 00:23:31.680
This is the car that's moving.

00:23:34.240 --> 00:23:36.580
It keeps going back and forth.

00:23:37.760 --> 00:23:41.320
Whenever the slope is positive, it's going forward.

00:23:41.520 --> 00:23:43.700
Remember the slope is negative, it's going backwards.

00:23:43.700 --> 00:23:47.380
okay we typically don't see this line but

00:23:47.380 --> 00:23:50.820
it's but it's moving like that

00:23:50.820 --> 00:23:55.320
and its velocity well this calculates it

00:23:55.320 --> 00:23:59.680
we'll learn later but the whole

00:23:59.680 --> 00:24:01.520
idea of a position function we're gonna deal with this

00:24:01.520 --> 00:24:05.940
a lot it's very important okay

00:24:05.940 --> 00:24:10.240
let's do another one How do I know

00:24:10.240 --> 00:24:12.700
this is a position function?

00:24:17.800 --> 00:24:21.400
The y value or the function value is in some

00:24:21.400 --> 00:24:25.140
distance measured in feet and the t value

00:24:25.140 --> 00:24:29.680
his measurements. So I have some ball, I'm

00:24:29.680 --> 00:24:33.380
throwing it up or something, dropping it, moving along a

00:24:33.380 --> 00:24:34.300
vertical line.

00:24:36.580 --> 00:24:39.880
So how do I find the average velocity over the

00:24:39.880 --> 00:24:40.900
first three seconds?

00:24:45.360 --> 00:24:47.180
Average velocity, we stuck around.

00:24:47.460 --> 00:24:48.110
Is?

00:24:50.620 --> 00:24:53.040
Slope of the secant line.

00:24:55.100 --> 00:24:56.520
Okay, well, let's shoot.

00:25:00.300 --> 00:25:01.900
Let me get a graph of this.

00:25:02.060 --> 00:25:02.710
Hold on.

00:25:04.180 --> 00:25:06.360
Okay, so this is the position function.

00:25:11.080 --> 00:25:13.240
We want to find the average velocity.

00:25:13.540 --> 00:25:16.260
This again is in seconds.

00:25:16.260 --> 00:25:18.740
This is in feet.

00:25:29.940 --> 00:25:32.120
We just have to find that slope of the second

00:25:32.120 --> 00:25:33.900
line. So average velocity.

00:25:35.920 --> 00:25:40.100
Let's see. From on 0 to

00:25:40.100 --> 00:25:41.000
3.

00:25:44.540 --> 00:25:47.140
It will be the change in y.

00:25:48.080 --> 00:25:50.000
How do I find the change in y?

00:25:55.640 --> 00:25:59.580
Y2 minus y1 or in function notation, be

00:25:59.580 --> 00:26:03.960
f of 3 minus f of

00:26:03.960 --> 00:26:08.440
0 over 3 minus 0.

00:26:13.380 --> 00:26:15.060
And what's our function?

00:26:15.200 --> 00:26:15.920
2t squared.

00:26:18.060 --> 00:26:22.400
2, 3 squared minus 2 times 0 squared.

00:26:23.460 --> 00:26:24.110
3.

00:26:25.940 --> 00:26:27.020
And

00:26:29.260 --> 00:26:33.780
18 over 3 that's 6 and what are the

00:26:33.780 --> 00:26:34.430
units?

00:26:38.220 --> 00:26:41.220
Each of these were in feet each of these were

00:26:41.220 --> 00:26:44.920
in seconds so it's feet per second

00:26:52.900 --> 00:26:53.550
Okay,

00:26:54.840 --> 00:26:56.660
our goal, I didn't write this down, our goal

00:26:56.660 --> 00:27:00.760
is estimate

00:27:00.760 --> 00:27:04.940
or find

00:27:04.940 --> 00:27:07.620
the instantaneous

00:27:12.240 --> 00:27:15.040
velocity at

00:27:15.040 --> 00:27:19.000
the 3 second mark.

00:27:20.700 --> 00:27:23.560
Was he going faster at three seconds than he was

00:27:23.560 --> 00:27:25.060
over the average?

00:27:27.420 --> 00:27:28.840
It's like three seconds higher.

00:27:30.540 --> 00:27:34.240
Sorry. So compared to the first three seconds, which he

00:27:34.240 --> 00:27:36.120
went six feet per second on average,

00:27:37.040 --> 00:27:38.900
is he going faster at the three second mark?

00:27:39.820 --> 00:27:40.880
Yes. How do we tell?

00:27:41.360 --> 00:27:42.420
Because the slope is steeper.

00:27:42.420 --> 00:27:46.880
the slope here is steeper okay definitely gaining

00:27:46.880 --> 00:27:51.480
speed gaining velocity so let's do it

00:27:51.480 --> 00:27:53.760
let's go from two to three

00:27:55.460 --> 00:27:59.900
So it's going to be f of 3 minus

00:27:59.900 --> 00:28:02.800
f of 2 over three minus

00:28:02.800 --> 00:28:05.220
two.

00:28:06.920 --> 00:28:07.960
18.

00:28:14.620 --> 00:28:16.040
10 feet per second.

00:28:18.340 --> 00:28:22.160
The first three seconds it averaged 6 feet per second.

00:28:22.580 --> 00:28:26.960
The last third of that it averaged 10 feet

00:28:26.960 --> 00:28:27.610
per second.

00:28:31.830 --> 00:28:33.660
So that would be the slope of this line.

00:28:36.600 --> 00:28:41.180
Our goal is to estimate the slope of the tangent

00:28:41.180 --> 00:28:41.830
line.

00:28:44.100 --> 00:28:45.420
So how are we going to do that?

00:28:50.600 --> 00:28:55.020
We're just going to pick intervals, very

00:28:55.020 --> 00:28:58.240
small intervals close to at 3.

00:28:58.240 --> 00:29:01.800
So let's go average velocity

00:29:01.800 --> 00:29:06.620
from

00:29:06.620 --> 00:29:10.740
2.9 seconds to 3

00:29:10.740 --> 00:29:11.390
seconds.

00:29:12.540 --> 00:29:14.220
This is what we're for

00:29:15.920 --> 00:29:16.640
11.8?

00:29:19.360 --> 00:29:20.820
In fact, let me show you how it goes.

00:29:22.160 --> 00:29:25.420
If you put the function into y1, let's have everybody

00:29:25.420 --> 00:29:26.420
do that just for fun.

00:29:28.640 --> 00:29:32.680
If you have a graphic calculator, press Y1, put the

00:29:32.680 --> 00:29:33.720
function in.

00:29:35.380 --> 00:29:39.660
And then instead of writing out 2 times something squared,

00:29:41.080 --> 00:29:43.600
I typed in Y1.

00:29:43.720 --> 00:29:46.160
It's kind of like our F, F of a number.

00:29:46.940 --> 00:29:50.780
To get to the Y1, you press alpha

00:29:50.780 --> 00:29:54.440
trate.

00:29:57.460 --> 00:30:00.100
And then you can either go to Y1, Y2.

00:30:00.260 --> 00:30:01.820
These are different function values.

00:30:07.620 --> 00:30:08.460
So what was it?

00:30:08.740 --> 00:30:12.240
11.8 feet per second.

00:30:20.320 --> 00:30:24.580
Okay, so the average over the last

00:30:24.580 --> 00:30:29.080
tenth of a second, which is the slope

00:30:29.080 --> 00:30:32.380
of the secant line over a very small period of

00:30:32.380 --> 00:30:33.060
time,

00:30:34.000 --> 00:30:37.220
is 11.8.

00:30:38.980 --> 00:30:43.340
Okay, so to get a better guess we just need

00:30:43.340 --> 00:30:44.860
to choose a smaller interval.

00:30:46.130 --> 00:30:48.860
So instead of 2.9, let's go

00:30:48.860 --> 00:30:51.320
average

00:30:51.320 --> 00:30:55.000
velocity from,

00:30:55.560 --> 00:30:59.580
let's go 2.999 seconds

00:30:59.580 --> 00:31:03.640
and 3 seconds. the last

00:31:03.640 --> 00:31:06.420
one thousandths of a second

00:31:06.420 --> 00:31:10.040
what was the average over that very very small period

00:31:10.040 --> 00:31:11.960
of time

00:31:13.660 --> 00:31:15.760
11.998.

00:31:17.460 --> 00:31:21.180
So the last one

00:31:21.180 --> 00:31:25.200
thousandths of a second he averaged, alright

00:31:25.200 --> 00:31:35.420
still

00:31:35.420 --> 00:31:37.380
an So the average

00:31:37.380 --> 00:31:40.560
velocity

00:31:40.560 --> 00:31:43.020
from 2

00:31:43.020 --> 00:31:47.560
.9999999 in 3

00:31:47.560 --> 00:31:48.540
seconds.

00:31:50.240 --> 00:31:53.360
Is that right?

00:31:53.360 --> 00:31:57.460
11.99998.

00:31:59.160 --> 00:32:03.120
So the last hundred thousandths of a second,

00:32:03.120 --> 00:32:07.020
the average velocity 11

00:32:07.020 --> 00:32:08.600
.99998.

00:32:10.340 --> 00:32:13.120
So what is our guess

00:32:13.120 --> 00:32:17.660
as to the

00:32:17.660 --> 00:32:20.180
instantaneous

00:32:20.180 --> 00:32:24.980
velocity

00:32:24.980 --> 00:32:29.420
at three seconds okay hopefully

00:32:29.420 --> 00:32:32.860
we see a pattern eleven point nine eight nine nine

00:32:32.860 --> 00:32:37.120
nine eight nine nine nine nine eight getting

00:32:37.120 --> 00:32:37.770
closer

00:32:37.620 --> 00:32:51.340
and

00:32:51.340 --> 00:32:55.160
closer and I'm gonna go right here.

00:33:01.160 --> 00:33:04.420
And then we are gonna learn how to very easily

00:33:04.420 --> 00:33:07.200
find it. But the whole point right now is,

00:33:08.100 --> 00:33:09.540
what are rates of change?

00:33:09.660 --> 00:33:10.420
What is the slope?

00:33:10.560 --> 00:33:12.200
And then how are we gonna estimate it when we

00:33:12.200 --> 00:33:16.680
don't have other

00:33:16.680 --> 00:33:19.280
options? We could have chosen a number just after three

00:33:19.280 --> 00:33:23.220
as well. Like if we could have gone between three

00:33:23.220 --> 00:33:27.720
and three point zero zero one

00:33:27.720 --> 00:33:29.200
or something seconds

00:33:29.200 --> 00:33:41.180
okay

00:33:41.180 --> 00:33:45.260
we're gonna get a number slightly greater than 12 let's

00:33:45.260 --> 00:33:47.800
see what it is Could somebody plug it in tell

00:33:47.800 --> 00:33:51.060
me where it is should be 12 point

00:33:52.760 --> 00:33:54.900
How many zeros did I say?

00:33:54.900 --> 00:33:58.220
zero one

00:33:58.220 --> 00:34:15.660
okay

00:34:15.660 --> 00:34:23.160
if

00:34:23.160 --> 00:34:25.460
you chose a number that gets closer to three but

00:34:25.460 --> 00:34:28.000
on the right side of three keep getting closer and

00:34:28.000 --> 00:34:32.360
closer you'll see there it's kept getting more zeros

00:34:32.360 --> 00:34:34.380
there and keep getting closer and closer to 12.

00:34:35.300 --> 00:34:38.700
Okay so for right now we're just going to estimate

00:34:38.700 --> 00:34:39.660
the average,

00:34:40.640 --> 00:34:44.420
sorry estimate the instantaneous rate of change, the

00:34:44.420 --> 00:34:48.100
velocity or whatever rate it might be at exactly a

00:34:48.100 --> 00:34:51.760
period of time and by choosing intervals that keep getting

00:34:51.760 --> 00:34:55.180
Smaller smaller smaller and closer to the value we're looking

00:34:55.180 --> 00:34:57.900
at or the time value we're looking at