Okay, a little review from
geometry.
What do we call a line that goes through
a circle and touches two different points?
What's that?
Secant line.
And what do
we call a
line that
touches it at one single point?
tangent line. Let's call it tangent line.
Why are these also the name of two of our
trig functions?
Is that just random or...
It does have something to do with secant and tangent
lines but we'll get to that some other day.
Um...
What's a word close to tangent that means
something similar?
Sorry, not a mathematical word.
An English word.
tangent line.
Maybe the circle looks something like a tangent line.
No, I'm looking for something else.
I look for the word tangible.
What does tangible mean?
You can touch it.
So it just touches it one time.
So I think that's where the word comes from.
Well the same is true for functions.
The
If a line crosses through a function at two different
points, it's called a secant line.
And if line touches
at one single point,
okay,
it's called the tangent line.
But it's a little bit harder to define.
For a circle, it's very simple.
A line that touches a circle just once is the
tangent line. You can't say the same here because this
line right here touches the function of one single point.
It's not tangent.
Okay, so it's a little harder to define what it
means to be tangent.
Okay, basically it has to be going in the exact
same direction at
that point.
I have a very similar slope to some line the
secant line goes through two points
We'll get to the actual definition of it.
Thank you. Bye.
Most of you have a basic understanding
What if we want to find the slope of the
secant line
What will we need?
The two ordered pairs, right?
Let's call this x1,
xy1. This
x2, y2. And
then the slope
is what for me?
y2 minus y1 over x2 minus x1.
Okay,
write the rise over the run.
What about the slope of the secant line?
Sorry, slope of the tangent line.
all we have is one single point
let's call it x1 y1
is there a nice formula for it?
all we have is one point
and we don't know the slope
that's if we want to find the equation of that
line yeah how do we even find the slope
is it just y1 over x1
definitely not
Okay, so who knows?
Okay, we're going to have to do some work and
eventually we're going to figure out how to do it.
But this is the major part of calculus,
uh, is how in the world are we going to
do this?
Slopes of lines are often called rates,
rate of change.
What is the definition of a rate?
Anybody remember from seventh grade?
What's the definition of a rate?
No thoughts?
Just the word rate.
For example,
uh... 50 miles per
hour is a rate.
Okay.
It is a ratio, but there's one more
thing.
So it's a ratio
of
two quantities.
units. Very important with different
units.
For example, like two-thirds is a ratio but it's
not a rate. Why is it not a rate?
There are no units or they were the same units
but two cups per
three
Gallons
That's not really good here, let's go three cups
of flour
two, three gallons
of water or something like
that. So as long as you have different units.
50 miles per hour, same thing as 50 miles
per one hour. You gotta have two
different units.
That's a rate.
Okay, going back here, the
slope here of a line is a rate.
Let's just say the slope of this line is
2
thirds.
Just making up some number.
I don't see units.
Why is it still a rate?
Because it's such a new y on the checkbook.
Yeah, x has some units.
We're studying writing down.
and y have some units I sit right down so
it's two y units
to three x
units
okay sometimes we're given a the units sometimes we're
not but it is always a rate okay and we
sometimes call a rate of change because it's how fast
the X is changing compared to how fast the y
is changing. Okay,
what are some examples of rates in real life?
I gave you one already, 50 miles per hour.
Discuss with somebody next to you, What are examples?
Giving an actual number and then the units.
Yeah, row one. Five cookies for ten dollars.
Okay!
Five...
cookies... to ten dollars.
Okay, row two, somebody give me one.
Or in other words, one tablespoon
per gallon.
Same thing. Okay. Is that how much you pee for
fish? I don't have fish.
You don't have fish?
Do you have fish?
No.
That's cool. Interesting. Somebody wrote three.
$20 an hour. Ooh, okay.
And what rate would
that be?
What's that? Your wage, often known as your
hourly rate.
Five ice cream scoop a minute.
That's the rate at which you can scoop ice cream.
Okay. I like that.
Do you eat a lot of ice cream?
and what would be that rate of?
population density
yeah I like that row 6
100 words per minute ooh I like that
Okay, and what rate is that the rate of?
Like, typing. I see you're typing speed.
Speed is always a type of rate.
Rate of change, same thing.
That's pretty fast.
Okay, anybody else? Give me some more examples.
throw something
out.
100
frames per
second and what's what's
that the rate of anybody?
So that'd be a frame rate how fast your TV
screen or a camera refreshes
or takes pictures so many frames per every
second.
Somebody else something.
$100,000 a year.
It's your yearly rate of pay.
Also your salary maybe.
You make a lot of money.
It's pretty good what else
we got
and what rate would that be interest rates
your interest rate okay depending How many years and how
many dollars you get a certain?
percentage that per year
like that anybody else This the type
of speed
These are all different speeds, but not about distance
But
We have
velocity and speed versus speed
uhh we all taken some of you have had physics
already?
so what's the difference between velocity and speed?
what's the term they usually use in physics?
well they typically use the word velocity
so what is the difference between velocity and speed?
The speed is velocity with no
direction.
So velocity is speed and
direction.
Okay.
And they typically deal with vectors and direction.
For us, in this class,
you only have two directions.
Either the positive direction...
...or the negative direction.
Okay.
Everything we do with velocity in this class, Whatever
is moving only moves on a single line.
It's either moving in the positive direction or it's moving
in the negative direction.
Everything in its class has to do with velocity and
speed. It's always on a straight line.
OK, this is very important.
And we either... So if you are negative
five miles per hour...
Okay, that means you're going backwards or in the negative
direction at 5 mph.
Okay, so speed is very simple.
it is the
absolute value of velocity.
Okay, this is very important.
So if your velocity is negative five miles per hour
What's your speed?
five miles per
Okay, if you're
32 feet per second
Feet per second
This is if this is your velocity, what's your speed?
32 feet direction. Okay, if it's positive you simply are
going to the right on a number line
everything here we move on a number line okay it
could be on a vertical number line up and down
horizontal number line just everything's on a number line and
velocity can be negative just being you're going backwards
Okay.
a position
function.
OK. So anytime you're given a function,
typically an equation,
and if the y variable is some distance
and the x variable is measured in any kind of
time period, it is a position function.
Okay.
How do I know this is a position function?
The x is in some time, measured in minutes, and
the y is some distance, in this case measured in
years.
Let's say this is me walking or something.
So this is Mr.
Whitman's position
at
any time
t in minutes.
Okay, what we don't see is some the line that
I'm walking on.
It's some other line that has nothing to do with
this graph, okay?
Okay at time zero where am I?
What's that?
and time, time, time, 10 meters.
So at time 0, which is the x-axis or
t-axis,
I'm exactly here. So I am right here
at time 0.
And at time,
oh, yeah.
These are all in meters.
And at time 10 minutes, where am I at?
I'm right here
Okay
Okay, and then I was during that ten minutes I
was moving
What was my average
Velocity
How do
we...
So average velocity is change total
change
in distance
divided by
my total change in
How did you get that?
I did it at 70 minus 10 because it started
at 10. And then 10 minus 0.
So that's 60 per...
70 minus 10 meters.
And 10 minus 0
minutes.
is 60 over 10
which is 6 and the units are
meters per minute
that's pretty slow walker by the
way.
Okay, which is exactly
the slope of this line.
Okay, your average velocity is
the same thing as the slope of the secant line.
Was I always the whole time
moving at 6 meters per minute?
How do we know?
The lines not linear
Okay, if this were my position function right here this
red line Then it would always be moving exactly six
meters per second
Let's say at one minute mark
oops Was it moving faster than six or slower
than six?
Faster and how can we tell?
Well if
If we found that slope with a secant line between
zero and one minute, it's way steeper, right?
So over that whole minute I was going way faster
But what about exactly at one minute
Well, what we could do is
Find the slope of two points real close to each
other
And let's say it was that.
You can keep getting closer and closer and closer.
And it's going to be a steeper slope than this
slope. So at one minute I was going faster.
Ideally we have to find the slope of the
tangent line.
The slope of the tangent line is your instantaneous
velocity. Okay, so for
any position function, the instantaneous
velocity
is the
slope of the
tangent line.
What we can't do is, well, I'll show you in
a second. I started right here.
At some point I was moving faster and then I
slowed down and almost stopped.
Because like around here, the slope of the tangent line
is practically zero.
So I, wherever I was at.
Nice little way down, I went real slow, real slow,
and then I sped up at the end.
Okay, it's hard to do it moving.
Let's uh... Okay, here's an example of a
position function. The function, I think it's x times sine
of x.
This is the car that's moving.
It keeps going back and forth.
Whenever the slope is positive, it's going forward.
Remember the slope is negative, it's going backwards.
okay we typically don't see this line but
it's but it's moving like that
and its velocity well this calculates it
we'll learn later but the whole
idea of a position function we're gonna deal with this
a lot it's very important okay
let's do another one How do I know
this is a position function?
The y value or the function value is in some
distance measured in feet and the t value
his measurements. So I have some ball, I'm
throwing it up or something, dropping it, moving along a
vertical line.
So how do I find the average velocity over the
first three seconds?
Average velocity, we stuck around.
Is?
Slope of the secant line.
Okay, well, let's shoot.
Let me get a graph of this.
Hold on.
Okay, so this is the position function.
We want to find the average velocity.
This again is in seconds.
This is in feet.
We just have to find that slope of the second
line. So average velocity.
Let's see. From on 0 to
3.
It will be the change in y.
How do I find the change in y?
Y2 minus y1 or in function notation, be
f of 3 minus f of
0 over 3 minus 0.
And what's our function?
2t squared.
2, 3 squared minus 2 times 0 squared.
3.
And
18 over 3 that's 6 and what are the
units?
Each of these were in feet each of these were
in seconds so it's feet per second
Okay,
our goal, I didn't write this down, our goal
is estimate
or find
the instantaneous
velocity at
the 3 second mark.
Was he going faster at three seconds than he was
over the average?
It's like three seconds higher.
Sorry. So compared to the first three seconds, which he
went six feet per second on average,
is he going faster at the three second mark?
Yes. How do we tell?
Because the slope is steeper.
the slope here is steeper okay definitely gaining
speed gaining velocity so let's do it
let's go from two to three
So it's going to be f of 3 minus
f of 2 over three minus
two.
18.
10 feet per second.
The first three seconds it averaged 6 feet per second.
The last third of that it averaged 10 feet
per second.
So that would be the slope of this line.
Our goal is to estimate the slope of the tangent
line.
So how are we going to do that?
We're just going to pick intervals, very
small intervals close to at 3.
So let's go average velocity
from
2.9 seconds to 3
seconds.
This is what we're for
11.8?
In fact, let me show you how it goes.
If you put the function into y1, let's have everybody
do that just for fun.
If you have a graphic calculator, press Y1, put the
function in.
And then instead of writing out 2 times something squared,
I typed in Y1.
It's kind of like our F, F of a number.
To get to the Y1, you press alpha
trate.
And then you can either go to Y1, Y2.
These are different function values.
So what was it?
11.8 feet per second.
Okay, so the average over the last
tenth of a second, which is the slope
of the secant line over a very small period of
time,
is 11.8.
Okay, so to get a better guess we just need
to choose a smaller interval.
So instead of 2.9, let's go
average
velocity from,
let's go 2.999 seconds
and 3 seconds. the last
one thousandths of a second
what was the average over that very very small period
of time
11.998.
So the last one
thousandths of a second he averaged, alright
still
an So the average
velocity
from 2
.9999999 in 3
seconds.
Is that right?
11.99998.
So the last hundred thousandths of a second,
the average velocity 11
.99998.
So what is our guess
as to the
instantaneous
velocity
at three seconds okay hopefully
we see a pattern eleven point nine eight nine nine
nine eight nine nine nine nine eight getting
closer
and
closer and I'm gonna go right here.
And then we are gonna learn how to very easily
find it. But the whole point right now is,
what are rates of change?
What is the slope?
And then how are we gonna estimate it when we
don't have other
options? We could have chosen a number just after three
as well. Like if we could have gone between three
and three point zero zero one
or something seconds
okay
we're gonna get a number slightly greater than 12 let's
see what it is Could somebody plug it in tell
me where it is should be 12 point
How many zeros did I say?
zero one
okay
if
you chose a number that gets closer to three but
on the right side of three keep getting closer and
closer you'll see there it's kept getting more zeros
there and keep getting closer and closer to 12.
Okay so for right now we're just going to estimate
the average,
sorry estimate the instantaneous rate of change, the
velocity or whatever rate it might be at exactly a
period of time and by choosing intervals that keep getting
Smaller smaller smaller and closer to the value we're looking
at or the time value we're looking at