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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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00:16:52,800 --> 00:16:56,660
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

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was moving

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What was my average

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Velocity

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00:18:50,740 --> 00:18:53,540
How do

287
00:18:53,540 --> 00:18:58,620
we...

288
00:18:58,620 --> 00:19:02,240
So average velocity is change total

289
00:19:02,240 --> 00:19:06,020
change

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00:19:06,020 --> 00:19:09,200
in distance

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00:19:09,200 --> 00:19:13,800
divided by

292
00:19:13,800 --> 00:19:17,520
my total change in

293
00:19:18,660 --> 00:19:21,860
How did you get that?

294
00:19:21,860 --> 00:19:24,820
I did it at 70 minus 10 because it started

295
00:19:24,820 --> 00:19:27,480
at 10. And then 10 minus 0.

296
00:19:28,600 --> 00:19:29,860
So that's 60 per...

297
00:19:32,780 --> 00:19:35,760
70 minus 10 meters.

298
00:19:38,060 --> 00:19:42,020
And 10 minus 0

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00:19:42,020 --> 00:19:43,860
minutes.

300
00:19:46,400 --> 00:19:50,480
is 60 over 10

301
00:19:50,480 --> 00:19:54,920
which is 6 and the units are

302
00:19:54,920 --> 00:19:59,160
meters per minute

303
00:19:59,160 --> 00:20:03,420
that's pretty slow walker by the

304
00:20:03,420 --> 00:20:04,070
way.

305
00:20:09,340 --> 00:20:12,560
Okay, which is exactly

306
00:20:19,860 --> 00:20:21,600
the slope of this line.

307
00:20:23,300 --> 00:20:27,840
Okay, your average velocity is

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00:20:27,840 --> 00:20:31,660
the same thing as the slope of the secant line.

309
00:20:33,360 --> 00:20:36,400
Was I always the whole time

310
00:20:36,400 --> 00:20:39,200
moving at 6 meters per minute?

311
00:20:40,280 --> 00:20:42,340
How do we know?

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00:20:46,780 --> 00:20:49,080
The lines not linear

313
00:20:49,980 --> 00:20:54,420
Okay, if this were my position function right here this

314
00:20:54,420 --> 00:20:58,880
red line Then it would always be moving exactly six

315
00:20:58,880 --> 00:21:00,120
meters per second

316
00:21:02,620 --> 00:21:05,620
Let's say at one minute mark

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00:21:06,720 --> 00:21:11,160
oops Was it moving faster than six or slower

318
00:21:11,160 --> 00:21:11,810
than six?

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00:21:14,240 --> 00:21:16,260
Faster and how can we tell?

320
00:21:17,960 --> 00:21:19,280
Well if

321
00:21:22,100 --> 00:21:25,560
If we found that slope with a secant line between

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00:21:25,560 --> 00:21:28,880
zero and one minute, it's way steeper, right?

323
00:21:29,400 --> 00:21:32,140
So over that whole minute I was going way faster

324
00:21:34,060 --> 00:21:36,560
But what about exactly at one minute

325
00:21:43,500 --> 00:21:45,780
Well, what we could do is

326
00:21:46,820 --> 00:21:50,700
Find the slope of two points real close to each

327
00:21:50,700 --> 00:21:51,350
other

328
00:21:53,440 --> 00:21:55,040
And let's say it was that.

329
00:21:59,120 --> 00:22:01,160
You can keep getting closer and closer and closer.

330
00:22:02,360 --> 00:22:05,940
And it's going to be a steeper slope than this

331
00:22:05,940 --> 00:22:08,460
slope. So at one minute I was going faster.

332
00:22:10,280 --> 00:22:14,340
Ideally we have to find the slope of the

333
00:22:14,340 --> 00:22:15,360
tangent line.

334
00:22:16,420 --> 00:22:20,260
The slope of the tangent line is your instantaneous

335
00:22:20,260 --> 00:22:24,860
velocity. Okay, so for

336
00:22:24,860 --> 00:22:28,240
any position function, the instantaneous

337
00:22:28,240 --> 00:22:33,160
velocity

338
00:22:33,160 --> 00:22:35,300
is the

339
00:22:35,300 --> 00:22:39,900
slope of the

340
00:22:39,900 --> 00:22:40,880
tangent line.

341
00:22:42,580 --> 00:22:46,600
What we can't do is, well, I'll show you in

342
00:22:46,600 --> 00:22:48,400
a second. I started right here.

343
00:22:49,200 --> 00:22:52,740
At some point I was moving faster and then I

344
00:22:52,740 --> 00:22:54,380
slowed down and almost stopped.

345
00:22:55,240 --> 00:22:59,520
Because like around here, the slope of the tangent line

346
00:22:59,520 --> 00:23:02,220
is practically zero.

347
00:23:03,740 --> 00:23:05,500
So I, wherever I was at.

348
00:23:08,620 --> 00:23:12,500
Nice little way down, I went real slow, real slow,

349
00:23:12,660 --> 00:23:14,860
and then I sped up at the end.

350
00:23:15,840 --> 00:23:18,160
Okay, it's hard to do it moving.

351
00:23:18,160 --> 00:23:22,580
Let's uh... Okay, here's an example of a

352
00:23:22,580 --> 00:23:26,360
position function. The function, I think it's x times sine

353
00:23:26,360 --> 00:23:27,240
of x.

354
00:23:28,660 --> 00:23:31,680
This is the car that's moving.

355
00:23:34,240 --> 00:23:36,580
It keeps going back and forth.

356
00:23:37,760 --> 00:23:41,320
Whenever the slope is positive, it's going forward.

357
00:23:41,520 --> 00:23:43,700
Remember the slope is negative, it's going backwards.

358
00:23:43,700 --> 00:23:47,380
okay we typically don't see this line but

359
00:23:47,380 --> 00:23:50,820
it's but it's moving like that

360
00:23:50,820 --> 00:23:55,320
and its velocity well this calculates it

361
00:23:55,320 --> 00:23:59,680
we'll learn later but the whole

362
00:23:59,680 --> 00:24:01,520
idea of a position function we're gonna deal with this

363
00:24:01,520 --> 00:24:05,940
a lot it's very important okay

364
00:24:05,940 --> 00:24:10,240
let's do another one How do I know

365
00:24:10,240 --> 00:24:12,700
this is a position function?

366
00:24:17,800 --> 00:24:21,400
The y value or the function value is in some

367
00:24:21,400 --> 00:24:25,140
distance measured in feet and the t value

368
00:24:25,140 --> 00:24:29,680
his measurements. So I have some ball, I'm

369
00:24:29,680 --> 00:24:33,380
throwing it up or something, dropping it, moving along a

370
00:24:33,380 --> 00:24:34,300
vertical line.

371
00:24:36,580 --> 00:24:39,880
So how do I find the average velocity over the

372
00:24:39,880 --> 00:24:40,900
first three seconds?

373
00:24:45,360 --> 00:24:47,180
Average velocity, we stuck around.

374
00:24:47,460 --> 00:24:48,110
Is?

375
00:24:50,620 --> 00:24:53,040
Slope of the secant line.

376
00:24:55,100 --> 00:24:56,520
Okay, well, let's shoot.

377
00:25:00,300 --> 00:25:01,900
Let me get a graph of this.

378
00:25:02,060 --> 00:25:02,710
Hold on.

379
00:25:04,180 --> 00:25:06,360
Okay, so this is the position function.

380
00:25:11,080 --> 00:25:13,240
We want to find the average velocity.

381
00:25:13,540 --> 00:25:16,260
This again is in seconds.

382
00:25:16,260 --> 00:25:18,740
This is in feet.

383
00:25:29,940 --> 00:25:32,120
We just have to find that slope of the second

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00:25:32,120 --> 00:25:33,900
line. So average velocity.

385
00:25:35,920 --> 00:25:40,100
Let's see. From on 0 to

386
00:25:40,100 --> 00:25:41,000
3.

387
00:25:44,540 --> 00:25:47,140
It will be the change in y.

388
00:25:48,080 --> 00:25:50,000
How do I find the change in y?

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00:25:55,640 --> 00:25:59,580
Y2 minus y1 or in function notation, be

390
00:25:59,580 --> 00:26:03,960
f of 3 minus f of

391
00:26:03,960 --> 00:26:08,440
0 over 3 minus 0.

392
00:26:13,380 --> 00:26:15,060
And what's our function?

393
00:26:15,200 --> 00:26:15,920
2t squared.

394
00:26:18,060 --> 00:26:22,400
2, 3 squared minus 2 times 0 squared.

395
00:26:23,460 --> 00:26:24,110
3.

396
00:26:25,940 --> 00:26:27,020
And

397
00:26:29,260 --> 00:26:33,780
18 over 3 that's 6 and what are the

398
00:26:33,780 --> 00:26:34,430
units?

399
00:26:38,220 --> 00:26:41,220
Each of these were in feet each of these were

400
00:26:41,220 --> 00:26:44,920
in seconds so it's feet per second

401
00:26:52,900 --> 00:26:53,550
Okay,

402
00:26:54,840 --> 00:26:56,660
our goal, I didn't write this down, our goal

403
00:26:56,660 --> 00:27:00,760
is estimate

404
00:27:00,760 --> 00:27:04,940
or find

405
00:27:04,940 --> 00:27:07,620
the instantaneous

406
00:27:12,240 --> 00:27:15,040
velocity at

407
00:27:15,040 --> 00:27:19,000
the 3 second mark.

408
00:27:20,700 --> 00:27:23,560
Was he going faster at three seconds than he was

409
00:27:23,560 --> 00:27:25,060
over the average?

410
00:27:27,420 --> 00:27:28,840
It's like three seconds higher.

411
00:27:30,540 --> 00:27:34,240
Sorry. So compared to the first three seconds, which he

412
00:27:34,240 --> 00:27:36,120
went six feet per second on average,

413
00:27:37,040 --> 00:27:38,900
is he going faster at the three second mark?

414
00:27:39,820 --> 00:27:40,880
Yes. How do we tell?

415
00:27:41,360 --> 00:27:42,420
Because the slope is steeper.

416
00:27:42,420 --> 00:27:46,880
the slope here is steeper okay definitely gaining

417
00:27:46,880 --> 00:27:51,480
speed gaining velocity so let's do it

418
00:27:51,480 --> 00:27:53,760
let's go from two to three

419
00:27:55,460 --> 00:27:59,900
So it's going to be f of 3 minus

420
00:27:59,900 --> 00:28:02,800
f of 2 over three minus

421
00:28:02,800 --> 00:28:05,220
two.

422
00:28:06,920 --> 00:28:07,960
18.

423
00:28:14,620 --> 00:28:16,040
10 feet per second.

424
00:28:18,340 --> 00:28:22,160
The first three seconds it averaged 6 feet per second.

425
00:28:22,580 --> 00:28:26,960
The last third of that it averaged 10 feet

426
00:28:26,960 --> 00:28:27,610
per second.

427
00:28:31,830 --> 00:28:33,660
So that would be the slope of this line.

428
00:28:36,600 --> 00:28:41,180
Our goal is to estimate the slope of the tangent

429
00:28:41,180 --> 00:28:41,830
line.

430
00:28:44,100 --> 00:28:45,420
So how are we going to do that?

431
00:28:50,600 --> 00:28:55,020
We're just going to pick intervals, very

432
00:28:55,020 --> 00:28:58,240
small intervals close to at 3.

433
00:28:58,240 --> 00:29:01,800
So let's go average velocity

434
00:29:01,800 --> 00:29:06,620
from

435
00:29:06,620 --> 00:29:10,740
2.9 seconds to 3

436
00:29:10,740 --> 00:29:11,390
seconds.

437
00:29:12,540 --> 00:29:14,220
This is what we're for

438
00:29:15,920 --> 00:29:16,640
11.8?

439
00:29:19,360 --> 00:29:20,820
In fact, let me show you how it goes.

440
00:29:22,160 --> 00:29:25,420
If you put the function into y1, let's have everybody

441
00:29:25,420 --> 00:29:26,420
do that just for fun.

442
00:29:28,640 --> 00:29:32,680
If you have a graphic calculator, press Y1, put the

443
00:29:32,680 --> 00:29:33,720
function in.

444
00:29:35,380 --> 00:29:39,660
And then instead of writing out 2 times something squared,

445
00:29:41,080 --> 00:29:43,600
I typed in Y1.

446
00:29:43,720 --> 00:29:46,160
It's kind of like our F, F of a number.

447
00:29:46,940 --> 00:29:50,780
To get to the Y1, you press alpha

448
00:29:50,780 --> 00:29:54,440
trate.

449
00:29:57,460 --> 00:30:00,100
And then you can either go to Y1, Y2.

450
00:30:00,260 --> 00:30:01,820
These are different function values.

451
00:30:07,620 --> 00:30:08,460
So what was it?

452
00:30:08,740 --> 00:30:12,240
11.8 feet per second.

453
00:30:20,320 --> 00:30:24,580
Okay, so the average over the last

454
00:30:24,580 --> 00:30:29,080
tenth of a second, which is the slope

455
00:30:29,080 --> 00:30:32,380
of the secant line over a very small period of

456
00:30:32,380 --> 00:30:33,060
time,

457
00:30:34,000 --> 00:30:37,220
is 11.8.

458
00:30:38,980 --> 00:30:43,340
Okay, so to get a better guess we just need

459
00:30:43,340 --> 00:30:44,860
to choose a smaller interval.

460
00:30:46,130 --> 00:30:48,860
So instead of 2.9, let's go

461
00:30:48,860 --> 00:30:51,320
average

462
00:30:51,320 --> 00:30:55,000
velocity from,

463
00:30:55,560 --> 00:30:59,580
let's go 2.999 seconds

464
00:30:59,580 --> 00:31:03,640
and 3 seconds. the last

465
00:31:03,640 --> 00:31:06,420
one thousandths of a second

466
00:31:06,420 --> 00:31:10,040
what was the average over that very very small period

467
00:31:10,040 --> 00:31:11,960
of time

468
00:31:13,660 --> 00:31:15,760
11.998.

469
00:31:17,460 --> 00:31:21,180
So the last one

470
00:31:21,180 --> 00:31:25,200
thousandths of a second he averaged, alright

471
00:31:25,200 --> 00:31:35,420
still

472
00:31:35,420 --> 00:31:37,380
an So the average

473
00:31:37,380 --> 00:31:40,560
velocity

474
00:31:40,560 --> 00:31:43,020
from 2

475
00:31:43,020 --> 00:31:47,560
.9999999 in 3

476
00:31:47,560 --> 00:31:48,540
seconds.

477
00:31:50,240 --> 00:31:53,360
Is that right?

478
00:31:53,360 --> 00:31:57,460
11.99998.

479
00:31:59,160 --> 00:32:03,120
So the last hundred thousandths of a second,

480
00:32:03,120 --> 00:32:07,020
the average velocity 11

481
00:32:07,020 --> 00:32:08,600
.99998.

482
00:32:10,340 --> 00:32:13,120
So what is our guess

483
00:32:13,120 --> 00:32:17,660
as to the

484
00:32:17,660 --> 00:32:20,180
instantaneous

485
00:32:20,180 --> 00:32:24,980
velocity

486
00:32:24,980 --> 00:32:29,420
at three seconds okay hopefully

487
00:32:29,420 --> 00:32:32,860
we see a pattern eleven point nine eight nine nine

488
00:32:32,860 --> 00:32:37,120
nine eight nine nine nine nine eight getting

489
00:32:37,120 --> 00:32:37,770
closer

490
00:32:37,620 --> 00:32:51,340
and

491
00:32:51,340 --> 00:32:55,160
closer and I'm gonna go right here.

492
00:33:01,160 --> 00:33:04,420
And then we are gonna learn how to very easily

493
00:33:04,420 --> 00:33:07,200
find it. But the whole point right now is,

494
00:33:08,100 --> 00:33:09,540
what are rates of change?

495
00:33:09,660 --> 00:33:10,420
What is the slope?

496
00:33:10,560 --> 00:33:12,200
And then how are we gonna estimate it when we

497
00:33:12,200 --> 00:33:16,680
don't have other

498
00:33:16,680 --> 00:33:19,280
options? We could have chosen a number just after three

499
00:33:19,280 --> 00:33:23,220
as well. Like if we could have gone between three

500
00:33:23,220 --> 00:33:27,720
and three point zero zero one

501
00:33:27,720 --> 00:33:29,200
or something seconds

502
00:33:29,200 --> 00:33:41,180
okay

503
00:33:41,180 --> 00:33:45,260
we're gonna get a number slightly greater than 12 let's

504
00:33:45,260 --> 00:33:47,800
see what it is Could somebody plug it in tell

505
00:33:47,800 --> 00:33:51,060
me where it is should be 12 point

506
00:33:52,760 --> 00:33:54,900
How many zeros did I say?

507
00:33:54,900 --> 00:33:58,220
zero one

508
00:33:58,220 --> 00:34:15,660
okay

509
00:34:15,660 --> 00:34:23,160
if

510
00:34:23,160 --> 00:34:25,460
you chose a number that gets closer to three but

511
00:34:25,460 --> 00:34:28,000
on the right side of three keep getting closer and

512
00:34:28,000 --> 00:34:32,360
closer you'll see there it's kept getting more zeros

513
00:34:32,360 --> 00:34:34,380
there and keep getting closer and closer to 12.

514
00:34:35,300 --> 00:34:38,700
Okay so for right now we're just going to estimate

515
00:34:38,700 --> 00:34:39,660
the average,

516
00:34:40,640 --> 00:34:44,420
sorry estimate the instantaneous rate of change, the

517
00:34:44,420 --> 00:34:48,100
velocity or whatever rate it might be at exactly a

518
00:34:48,100 --> 00:34:51,760
period of time and by choosing intervals that keep getting

519
00:34:51,760 --> 00:34:55,180
Smaller smaller smaller and closer to the value we're looking

520
00:34:55,180 --> 00:34:57,900
at or the time value we're looking at