Calculus — Rates of Change and Instantaneous Velocity
Corrected lesson transcript


## 0:00 — Rates of Change — Secant and Tangent Lines

[0:04] Okay, a little review from geometry.
[0:16] What do we call a line that goes through a circle and touches two different points?
[0:26] What's that?
[0:28] Secant line.
[0:35] And what do we call a line that touches it at one single point?
[0:47] tangent line.
[0:47] Let's call it tangent line.
[0:55] Why are these also the name of two of our trig functions? Is that just random or...
[1:08] It does have something to do with secant and tangent lines but we'll get to that some other day.
[1:17] Um...
[1:24] What's a word close to tangent that means something similar?
[1:33] Sorry, not a mathematical word. An English word.
[1:37] tangent line.
[1:39] tangent line. Maybe the circle looks something like a tangent line. No, I'm looking for something
[1:44] else. I look for the word tangible. What does tangible mean? You can touch it. So it just
[2:00] touches it one time. So I think that's where the word comes from. Well the same
[2:09] is true for functions.
[2:11] The

## 2:14 — Secant and Tangent Lines on a Function

[2:14] If a line crosses through a function at two different points, it's called a
[2:20] secant line.
[2:22] And if line touches at one single point, okay,
[2:41] it's called the tangent line.
[2:52] But it's a little bit harder to define.
[2:57] For a circle, it's very simple.
[3:00] A line that touches a circle just once is the tangent line.
[3:04] You can't say the same here because this line right here touches the function of one single point.
[3:11] It's not tangent.
[3:14] Okay, so it's a little harder to define what it means to be tangent.
[3:18] Okay, basically it has to be going in the exact same direction at that point.
[3:27] I have a very similar slope to some line the secant line goes through two points
[3:34] We'll get to the actual definition of it. Thank you. Bye. Most of you have a basic understanding
[3:41] What if we want to find the slope of the secant line
[3:45] What will we need?
[3:49] The two ordered pairs, right?
[3:51] Let's call this x1, xy1.
[3:56] This x2, y2.
[4:00] And then the slope is what for me? y2 minus y1 over x2 minus x1.
[4:15] Okay, write the rise over the run.
[4:21] What about the slope of the secant line?
[4:31] Sorry, slope of the tangent line.
[4:42] all we have is one single point
[4:47] let's call it x1 y1
[4:54] is there a nice formula for it?
[4:59] all we have is one point
[5:04] and we don't know the slope
[5:07] that's if we want to find the equation of that line
[5:10] yeah how do we even find the slope
[5:16] is it just y1 over x1
[5:20] definitely not
[5:22] definitely not
[5:25] Okay, so who knows?
[5:30] Okay, we're going to have to do some work and eventually we're going to figure out how
[5:34] to do it.
[5:35] But this is the major part of calculus, uh, is how in the world are we going to do this?

## 5:44 — Rates and Rates of Change

[5:44] Slopes of lines are often called rates, rate of change.
[5:55] What is the definition of a rate?
[6:01] Anybody remember from seventh grade?
[6:05] What's the definition of a rate?
[6:08] No thoughts?
[6:17] Just the word rate.
[6:22] For example, uh...
[6:26] 50 miles per hour is a rate.
[6:32] Okay.
[6:34] It is a ratio, but there's one more thing.
[6:43] So it's a ratio of two quantities.
[7:00] units.
[7:00] Very important with different units.
[7:09] For example, like two-thirds is a ratio but it's not a rate.
[7:13] Why is it not a rate?
[7:16] There are no units or they were the same units but two cups per three
[7:27] Gallons
[7:33] That's not really good here, let's go three cups of flour
[7:41] two, three gallons of water or something like that.
[7:50] So as long as you have different units.
[7:52] 50 miles per hour, same thing as 50 miles per one hour.
[7:59] You gotta have two different units.
[8:02] That's a rate.
[8:04] Okay, going back here, the slope here of a line is a rate.
[8:14] Let's just say the slope of this line is 2 thirds.
[8:24] Just making up some number.
[8:28] I don't see units.
[8:30] Why is it still a rate?
[8:35] Because it's such a new y on the checkbook.
[8:37] Yeah, x has some units.
[8:40] We're studying writing down.
[8:41] and y have some units I sit right down so it's two y units to three x units
[8:55] okay sometimes we're given a the units sometimes we're not but it is always a
[9:01] rate okay and we sometimes call a rate of change because it's how fast the X is
[9:07] changing compared to how fast the y is changing.
[9:11] Okay, what are some examples of rates in real life?
[9:28] I gave you one already, 50 miles per hour.
[9:35] Discuss with somebody next to you,
[9:36] What are examples?
[9:39] Giving an actual number and then the units.
[9:42] Yeah, row one.
[9:42] Five cookies for ten dollars.
[9:44] Okay!
[9:49] Five...
[9:52] cookies...
[9:54] to ten dollars.
[9:57] Okay, row two, somebody give me one.
[10:00] Or in other words, one tablespoon per gallon.
[10:06] Same thing. Okay.
[10:08] Is that how much you pee for fish?
[10:09] I don't have fish.
[10:10] You don't have fish?
[10:11] Do you have fish? No.
[10:14] That's cool. Interesting.
[10:15] Somebody wrote three. $20 an hour. Ooh, okay. And what rate would that be? What's that?
[10:36] Your wage, often known as your hourly rate.
[10:40] Five ice cream scoop a minute.
[10:44] That's the rate at which you can scoop ice cream. Okay. I like that. Do you eat a lot of ice cream?
[10:51] and what would be that rate of?
[10:57] population density
[10:59] yeah I like that
[11:01] row 6
[11:10] 100 words per minute
[11:12] ooh I like that
[11:17] Okay, and what rate is that the rate of?
[11:21] Like, typing.
[11:22] I see you're typing speed.
[11:24] Speed is always a type of rate.
[11:29] Rate of change, same thing.
[11:32] That's pretty fast.
[11:37] Okay, anybody else?
[11:38] Give me some more examples.
[11:39] throw something out. 100 frames per second And what is that a rate of?
[11:59] So that'd be a frame rate how fast your TV screen or a camera refreshes or takes
[12:07] pictures so many frames per every second. Somebody else something. $100,000 a year.
[12:27] It's your yearly rate of pay. Also your salary maybe. You make a lot of money. It's pretty
[12:33] good what else we got
[12:39] and what rate would that be interest rates your interest rate okay depending
[12:46] How many years and how many dollars you get a certain?
[12:51] percentage that per year
[12:53] like that
[12:54] anybody else
[12:56] This the type of speed
[12:59] These are all different speeds, but not about distance
[13:05] But
[13:08] We have

## 13:13 — Velocity Versus Speed

[13:13] velocity and speed versus speed
[13:18] uhh we all taken some of you have had physics already?
[13:23] so what's the difference between velocity and speed?
[13:26] what's the term they usually use in physics?
[13:32] well they typically use the word velocity
[13:36] so what is the difference between velocity and speed?
[13:39] The speed is velocity with no direction.
[13:47] So velocity is speed and direction.
[13:56] Okay.
[14:00] And they typically deal with vectors and direction.
[14:05] For us, in this class, you only have two directions.
[14:11] Either the positive direction...
[14:18] ...or the negative direction.
[14:27] Okay.
[14:29] Everything we do with velocity in this class,
[14:32] Whatever is moving only moves on a single line.
[14:37] It's either moving in the positive direction or it's moving in the negative direction.
[14:43] Everything in its class has to do with velocity and speed.
[14:47] It's always on a straight line.
[14:50] OK, this is very important. And we either...
[14:53] So if you are negative five miles per hour...
[15:02] Okay, that means you're going backwards or in the negative direction at 5 mph.
[15:09] Okay, so speed is very simple.
[15:12] it is the absolute value of velocity.
[15:20] Okay, this is very important. So if your velocity is negative five miles per hour
[15:27] What's your speed?
[15:29] five miles per
[15:31] Okay, if you're
[15:35] 32 feet per second
[15:41] Feet per second
[15:45] This is if this is your velocity, what's your speed?
[15:50] 32 feet direction. Okay, if it's positive you simply are going to the right
[15:57] on a number line everything here we move on a number line okay it could be on a vertical
[16:02] number line up and down horizontal number line just everything's on a number line
[16:08] and velocity can be negative just being you're going backwards
[16:12] Okay.
[16:16] a position function.
[16:23] OK.

## 16:24 — Position Functions

[16:24] So anytime you're given a function, typically an equation, and if the y variable is some
[16:33] distance and the x variable is measured in any kind of time period, it is a position
[16:41] function.
[16:41] Okay.
[16:45] How do I know this is a position function?
[16:52] The x is in some time, measured in minutes, and the y is some distance, in this case measured
[16:58] in meters.
[17:00] Let's say this is me walking or something.
[17:04] So this is Mr. Wittman’s position at any time t in minutes.
[17:33] Okay, what we don't see is some the line that I'm walking on.
[17:38] It's some other line that has nothing to do with this graph, okay?
[17:45] Okay at time zero where am I?
[17:51] What's that?
[17:53] and time, time, time, 10 meters.
[17:57] So at time 0, which is the x-axis or t-axis,
[18:01] I'm exactly here.
[18:03] So I am right here at time 0.
[18:10] And at time, oh, yeah.
[18:16] These are all in meters.
[18:20] And at time 10 minutes, where am I at?
[18:24] I'm right here
[18:27] Okay
[18:30] Okay, and then I was during that ten minutes I was moving

## 18:36 — Average Velocity as a Secant Slope

[18:36] What was my average
[18:42] Velocity
[18:50] How do we...
[18:58] So average velocity is change total change in distance divided by my total change in
[19:18] How did you get that?
[19:21] I did it at 70 minus 10 because it started at 10.
[19:25] And then 10 minus 0.
[19:28] So that's 60 per...
[19:32] 70 minus 10 meters.
[19:38] And 10 minus 0 minutes.
[19:46] is 60 over 10 which is 6 and the units are meters per minute that's pretty slow walker
[20:03] by the way.
[20:09] Okay, which is exactly
[20:20] the slope of this line.
[20:23] Okay, your average velocity is the same thing as the slope of the secant line.
[20:33] Was I always the whole time moving at 6 meters per minute?
[20:40] How do we know?
[20:46] The lines not linear
[20:49] Okay, if this were my position function right here this red line
[20:55] Then it would always be moving exactly six meters per second
[21:02] Let's say at one minute mark
[21:06] oops
[21:08] Was it moving faster than six or slower than six?
[21:14] Faster and how can we tell?
[21:17] Well if
[21:22] If we found that slope with a secant line between zero and one minute, it's way steeper, right?
[21:29] So over that whole minute I was going way faster
[21:34] But what about exactly at one minute

## 21:43 — Instantaneous Velocity as a Tangent Slope

[21:43] Well, what we could do is
[21:46] Find the slope of two points real close to each other
[21:53] And let's say it was that.
[21:59] You can keep getting closer and closer and closer.
[22:02] And it's going to be a steeper slope than this slope.
[22:06] So at one minute I was going faster.
[22:10] Ideally we have to find the slope of the tangent line.
[22:16] The slope of the tangent line is your instantaneous velocity.
[22:21] Okay, so for any position function, the instantaneous velocity is the slope of the tangent line.
[22:42] What we can't do is, well, I'll show you in a second.
[22:47] I started right here.
[22:49] At some point I was moving faster and then I slowed down and almost stopped.
[22:55] Because like around here, the slope of the tangent line is practically zero.
[23:03] So I, wherever I was at.
[23:08] Nice little way down, I went real slow, real slow, and then I sped up at the end.
[23:15] Okay, it's hard to do it moving.
[23:18] Let's uh...
[23:20] Okay, here's an example of a position function.
[23:24] The function is x times sine of x.
[23:28] This is the car that's moving.
[23:34] It keeps going back and forth.
[23:37] Whenever the slope is positive, it's going forward.
[23:41] Remember the slope is negative, it's going backwards.
[23:43] okay we typically don't see this line but it's but it's moving like that and
[23:52] its velocity well this calculates it we'll learn later but the whole idea of
[24:00] a position function we're gonna deal with this a lot it's very important okay
[24:05] let's do another one

## 24:08 — Worked Example: f(t) = 2t²

[24:08] How do I know this is a position function?
[24:17] The y value or the function value is in some distance measured in feet and the t value
[24:25] his measurements. So I have some ball, I'm throwing it up or something, dropping it,
[24:32] moving along a vertical line. So how do I find the average velocity over the first
[24:40] three seconds? Average velocity, we stuck around. Is? Slope of the secant line.
[24:55] Okay, well, let's shoot.
[25:00] Let me get a graph of this. Hold on.
[25:04] Okay, so this is the position function.
[25:11] We want to find the average velocity.
[25:13] This again is in seconds.
[25:16] This is in feet. We just have to find that slope of the second line.
[25:32] So average velocity.
[25:35] Let's see.
[25:37] From on 0 to 3.
[25:44] It will be the change in y. How do I find the change in y?
[25:55] Y2 minus y1 or in function notation,
[25:58] be f of 3 minus f of 0 over 3 minus 0.
[26:13] And what's our function? 2t squared. 2, 3 squared minus 2 times 0 squared. 3. And
[26:29] 18 over 3 that's 6 and what are the units?
[26:38] Each of these were in feet each of these were in seconds so it's feet per second
[26:52] Okay, our goal, I didn't write this down, our goal is estimate or find the instantaneous
[27:12] velocity at the 3 second mark.
[27:20] Was he going faster at three seconds than he was over the average?
[27:27] It's like three seconds higher.
[27:30] Sorry.
[27:31] So compared to the first three seconds, which he went six feet per second on average, is
[27:37] he going faster at the three second mark?
[27:39] Yes.
[27:40] How do we tell?
[27:41] Because the slope is steeper.
[27:42] the slope here is steeper okay definitely gaining speed gaining velocity so let's
[27:51] do it let's go from two to three
[27:55] So it's going to be f of 3 minus f of 2 over three minus two.
[28:06] 18.
[28:14] 10 feet per second.
[28:18] The first three seconds it averaged 6 feet per second.
[28:22] The last third of that it averaged 10 feet per second.
[28:32] So that would be the slope of this line.
[28:36] Our goal is to estimate the slope of the tangent line.
[28:44] So how are we going to do that?

## 28:50 — Shrinking the Time Interval

[28:50] We're just going to pick intervals, very small intervals close to at 3.
[28:58] So let's go average velocity from 2.9 seconds to 3 seconds. This is what we're
[29:13] for
[29:15] 11.8?
[29:19] In fact, let me show you how it goes.
[29:22] If you put the function into y1, let's have everybody do that just for fun.
[29:28] If you have a graphic calculator, press Y1, put the function in.
[29:35] And then instead of writing out 2 times something squared, I typed in Y1.
[29:43] It's kind of like our F, F of a number.
[29:46] To get to the Y1, you press alpha trate.
[29:57] And then you can either go to Y1, Y2.
[30:00] These are different function values.
[30:07] So what was it?
[30:08] 11.8 feet per second.
[30:20] Okay, so the average over the last tenth of a second, which is the slope of the secant
[30:30] line over a very small period of time, is 11.8.
[30:38] Okay, so to get a better guess we just need to choose a smaller interval.
[30:46] So instead of 2.9, let's go average velocity from, let's go 2.999 seconds and 3 seconds.
[31:02] the last one thousandths of a second what was the average over that very very
[31:09] small period of time
[31:13] 11.998.
[31:17] So the last one thousandths of a second he averaged, alright still an
[31:35] So the average velocity from 2.9999999 in 3 seconds.
[31:50] Is that right?
[31:53] 11.99998.

## 31:59 — Estimating the Limiting Velocity

[31:59] So the last hundred thousandths of a second,
[32:03] the average velocity 11.99998. So what is our guess as to the instantaneous velocity
[32:24] at three seconds okay hopefully we see a pattern eleven point nine eight nine
[32:32] nine nine eight nine nine nine nine eight getting closer and closer and
[32:54] I'm gonna go right here.
[33:01] And then we are gonna learn how to very easily find it.
[33:05] But the whole point right now is,
[33:08] what are rates of change?
[33:09] What is the slope?
[33:10] And then how are we gonna estimate it
[33:11] when we don't have other options?
[33:17] We could have chosen a number just after three as well.
[33:20] Like if we could have gone between three
[33:23] and three point zero zero one or something seconds okay we're gonna get a
[33:42] number slightly greater than 12 let's see what it is
[33:46] Could somebody plug it in tell me where it is should be 12 point
[33:52] How many zeros did I say?
[33:54] zero one okay if you chose a number that gets closer to
[34:24] three but on the right side of three keep getting closer and closer you'll see there it's kept
[34:30] getting more zeros there and keep getting closer and closer to 12. Okay so for right now we're just
[34:37] going to estimate the average, sorry estimate the instantaneous rate of change, the velocity or
[34:45] whatever rate it might be at exactly a period of time and by choosing intervals that keep getting
[34:51] Smaller smaller smaller and closer to the value we're looking at or the time value we're looking at