The paradox of the derivative | Chapter 2, Essence of calculus

Summary

This script delves into the concept of derivatives, emphasizing their subtleties and potential paradoxes. It clarifies that while derivatives are often described as 'instantaneous rates of change,' this phrase is misleading, as true change occurs over time. Using a car's motion as an analogy, the script illustrates how velocity, or the derivative of distance over time, is calculated. It explains that velocity at a single moment is nonsensical without reference to a time interval. The true mathematical derivative is the limit of the ratio of small changes in distance over time as this interval approaches zero, equating to the slope of the tangent line at a point on the graph. The script challenges the audience to think critically about this concept, hinting at deeper exploration in future videos.

Takeaways

• 📚 The derivative is a fundamental concept in calculus that represents the rate of change of a function at a certain point, often referred to as the instantaneous rate of change.
• 🤔 The term 'instantaneous rate of change' is paradoxical because true change occurs over time, and at a single instant, there is no change.
• 🚗 The script uses the analogy of a car's motion to illustrate the concept of the derivative, where the car's velocity at any point is related to the slope of the distance-time graph at that point.
• 📈 The velocity function, representing the car's speed at different times, is derived from the distance function, which shows how far the car has traveled over time.
• 🔍 Velocity at a single moment is conceptually challenging because velocity is defined as the change in distance over time, requiring two points in time for calculation.
• 🔬 In reality, a car's speedometer measures speed over an infinitesimally small time interval, effectively sidestepping the paradox of instantaneous velocity.
• 📉 The derivative is mathematically defined as the limit of the ratio of the infinitesimal change in the function's value (ds) to the infinitesimal change in the input (dt) as dt approaches zero.
• 📐 The concept of a derivative provides a way to discuss the rate of change at a single point without directly addressing the paradox of change in an instant.
• 🧮 Calculating a derivative involves algebraic manipulation, which can be simplified by considering the limit as dt approaches zero, often leading to a cleaner, more understandable expression.
• 🚦 The script highlights the importance of understanding derivatives not as instantaneous values but as the best linear approximation of a rate of change near a point.

Q & A

What is the primary goal of the script?

The primary goal of the script is to explain what a derivative is, with an emphasis on understanding the subtleties and avoiding potential paradoxes associated with the concept.

Why is the phrase 'instantaneous rate of change' considered an oxymoron?

The phrase 'instantaneous rate of change' is considered an oxymoron because change implies a process that happens over time, involving separate points, whereas an instant does not have duration, leaving no room for change to occur.

How does the script illustrate the concept of a derivative?

The script uses the example of a car's motion from point A to point B over 10 seconds, graphing distance against time, to illustrate the concept of a derivative. It explains how the steepness of the curve (velocity) is related to the rate of change in distance over time.

What is the relationship between the distance function and the velocity function as described in the script?

The distance function and the velocity function are related such that the velocity at a given moment is the rate of change of the distance with respect to time. The script explains that the velocity function can be derived from the distance function by considering small changes in distance over small changes in time.

Why does the script suggest that velocity at a single moment doesn't make sense?

The script suggests that velocity at a single moment doesn't make sense because velocity is defined as the change in distance over time, which inherently requires two separate points in time for comparison.

How does a car's speedometer handle the paradox of measuring velocity at a single instant?

A car's speedometer handles the paradox by measuring the distance traveled over a very small, but non-zero, amount of time (dt), and then computing the speed as the change in distance (ds) divided by this small time interval.

What does the script mean by 'the derivative is not the ratio ds/dt for a specific choice of dt'?

The script means that in pure mathematics, the derivative is not just the ratio of the change in distance (ds) to the change in time (dt) for a specific, finite dt. Instead, it is the limit of this ratio as dt approaches zero, which represents the slope of the tangent line to the function at a specific point.

How does the script resolve the paradox of change in an instant?

The script resolves the paradox by introducing the concept of a limit, where the derivative is defined as the limit of the ratio ds/dt as dt approaches zero. This approach allows for discussing the rate of change at a single point without directly implying change occurs at an instant.

What is the significance of the derivative being the slope of the tangent line to a graph?

The significance of the derivative being the slope of the tangent line to a graph is that it provides a precise, mathematical way to describe the rate of change at a specific point on a curve, which is a fundamental concept in calculus and has wide applications in science and engineering.

Why does the script emphasize that the derivative should not be thought of as an instantaneous rate of change?

The script emphasizes that the derivative should not be thought of as an instantaneous rate of change because the concept of change at a single instant is paradoxical. Instead, the derivative should be understood as the best constant approximation for the rate of change around a point.

How does the script use the example of a car starting at time t equals 0 to illustrate a paradox?

The script uses the example to illustrate a paradox by asking whether the car is moving at time t equals 0. According to the derivative, the velocity (and thus the rate of change) is zero, suggesting it's not moving, yet the car must start moving at some point, leading to a conceptual conflict about the nature of instantaneous velocity.

Outlines

00:00

📘 Introduction to Derivatives

The paragraph introduces the concept of derivatives, emphasizing the subtleties and potential paradoxes associated with the topic. It challenges the common but misleading notion of an 'instantaneous rate of change' by highlighting the contradiction in the idea of change happening at a single instant. The paragraph sets up the context for understanding derivatives by using the example of a car's motion from point A to point B over 10 seconds. It explains the graphical representation of distance versus time and the relationship between the distance function and velocity, leading to a deeper exploration of how velocity is derived from the distance function.

05:03

🔍 Understanding Velocity and the Derivative

This section delves into the concept of velocity and its mathematical representation. It explains that velocity, while intuitively understood as the speed shown on a car's speedometer, is not something that can be measured at an exact instant. Instead, velocity is calculated over an infinitesimally small interval of time. The paragraph introduces the mathematical notation for this concept, using 'ds/dt' to represent the change in distance over the change in time. It also discusses how a computer might calculate velocity from a distance-time graph by considering small time intervals and the corresponding changes in distance, leading to the concept of the derivative as the limit of 'ds/dt' as 'dt' approaches zero.

10:06

📐 The Tangent Line and Derivative Notation

The paragraph further clarifies the concept of the derivative by likening it to the slope of the tangent line to a curve at a specific point. It explains that the derivative is not merely the slope between two points on a graph, but rather the limit of this slope as the interval between the points shrinks to zero. The paragraph also addresses the mathematical notation used for derivatives, emphasizing that 'ds/dt' is not a traditional fraction but represents the limit of the ratio as 'dt' approaches zero. An example is provided to illustrate how the derivative simplifies the calculation process, showing that the derivative of 't cubed' is '3t squared', which is a straightforward result despite the complexity of the initial concept.

15:07

🤔 Paradox of Instantaneous Velocity

The final paragraph addresses a paradox that arises when considering the derivative as an instantaneous rate of change. It uses the example of a car's motion at time t equals zero to question whether the car is moving at that exact moment. The discussion highlights the limitations of the concept of instantaneous velocity and suggests that the derivative should be thought of as the best constant approximation for the rate of change around a point rather than an exact measure at an instant. The paragraph concludes by emphasizing the importance of understanding the derivative as a tool for approximating rates of change rather than as a measure of change at a single point in time.

Keywords

💡Derivative

A derivative in calculus represents the rate at which a function changes. It is the ratio of the infinitesimal increment of the output over the infinitesimal increment of the input. In the video, the derivative is used to describe the velocity of a car as a function of time, emphasizing the subtlety of defining instantaneous rate of change.

💡Instantaneous Rate of Change

This concept refers to the rate of change at a specific instant, which is paradoxical because change inherently involves time intervals. The video script uses this term to highlight the oxymoronic nature of discussing change at a single point in time, which is a central theme in understanding derivatives.

💡Paradox

A paradox is a statement that despite sound reasoning from acceptable premises results in an apparently self-contradictory or logically unacceptable conclusion. The video discusses the paradox of defining velocity at a single instant, which is a key point in understanding the nuances of derivatives.

💡Velocity

Velocity is a measure of the rate of change of position with respect to time. In the script, velocity is used as an example to illustrate the concept of a derivative, showing how it can be calculated as the slope of the tangent line to a curve representing distance traveled over time.

💡Distance Function

A distance function, often denoted as s(t) in the script, represents the total distance traveled by an object as a function of time. It is used to model the motion of the car in the video, with the derivative of this function giving the velocity at any given time.

💡Tangent

A tangent to a curve at a point is a straight line that touches the curve at that point. The video explains that the derivative of a function at a point is the slope of the tangent line to the curve of the function at that point, which provides a visual way to understand derivatives.

💡Limit

In calculus, a limit is the value that a function or sequence approaches as the input approaches a certain value. The script discusses how the derivative is defined in terms of limits, specifically as the limit of the ratio of the change in the function to the change in the input as this change approaches zero.

💡Rate of Change

The rate of change is the measure of how quickly a quantity changes with respect to another quantity. The video uses the rate of change to explain the concept of velocity and how it can be approximated by the derivative, even though the concept of instantaneous rate of change is paradoxical.

💡Oxymoron

An oxymoron is a figure of speech in which opposite or contradictory ideas are combined to create a new meaning. The script uses the term 'instantaneous rate of change' as an example of an oxymoron to emphasize the conceptual difficulty in defining change at a single instant.

💡Slope

Slope in mathematics is a measure of the steepness of a line. In the context of the video, the slope of the line on a graph representing distance versus time is used to explain the concept of velocity and how it relates to the derivative of the distance function.

💡Function

A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video uses functions to model various physical quantities such as distance and velocity, and how their derivatives relate to rates of change.

Highlights

The goal is to explain what a derivative is and to appreciate the subtleties and paradoxes involved.

The derivative is often described as an instantaneous rate of change, which is paradoxical since change occurs between points in time.

The fathers of calculus were praised for capturing the idea of instantaneous rate of change with a sensible mathematical concept.

A car's motion from point A to point B is used as a central example to illustrate the concept of the derivative.

The distance-time graph is introduced to represent the car's motion, with distance on the vertical axis and time on the horizontal axis.

The velocity of the car is represented as a bump curve on a graph, showing its variation over time.

Velocity is related to the distance vs. time function, and the derivative helps understand this relationship.

Velocity at a single moment is paradoxical because velocity requires comparison between two points in time.

The car's speedometer measures speed over a very small time interval, effectively sidestepping the paradox.

The derivative is introduced as the limit of the ratio ds/dt as dt approaches 0, representing the slope of the tangent line at a point.

The derivative is not about an infinitely small change but rather what the ratio approaches as dt approaches 0.

The derivative is best thought of as the best constant approximation for the rate of change around a point, not an instantaneous rate.

The notation for the derivative uses 'd' to indicate the intention to consider what happens as dt approaches 0.

An example is given where the derivative of t cubed is computed, simplifying to 3t squared, showcasing the power of calculus.

The derivative simplifies complex expressions by allowing us to ignore terms that become negligible as dt approaches 0.

A paradox is presented regarding the car's motion at t equals 0, questioning whether the car is moving or not, highlighting the limitations of the concept of instantaneous velocity.

The derivative is a powerful tool in calculus that provides a constant approximation for rate of change, despite the paradoxes involved.

Transcripts

The goal here is simple, explain what a derivative is.

The thing is though, there's some subtlety to this topic,

and a lot of potential for paradoxes if you're not careful.

So a secondary goal is that you have an appreciation

for what those paradoxes are and how to avoid them.

You see, it's common for people to say that the derivative measures an instantaneous

rate of change, but when you think about it, that phrase is actually an oxymoron.

Change is something that happens between separate points in time,

and when you blind yourself to all but just a single instant,

there's not really any room for change.

You'll see what I mean more as we get into it,

but when you appreciate that a phrase like instantaneous rate of change is actually

nonsense, I think it makes you appreciate just how clever the fathers of calculus

were in capturing the idea that phrase is meant to evoke,

but with a perfectly sensible piece of math, the derivative.

As our central example, I want you to imagine a car that starts at some point A,

speeds up, and then slows down to a stop at some point B 100 meters away,

and let's say it all happens over the course of 10 seconds.

That's the setup to have in mind as we lay out what the derivative is.

Well, we could graph this motion, letting the vertical axis represent the

distance traveled, and the horizontal axis represent time, so at each time t,

represented with a point somewhere on the horizontal axis,

the height of the graph tells us how far the car has traveled in total after

that amount of time.

It's pretty common to name a distance function like this s of t.

I would use the letter d for distance, but that

guy already has another full time job in calculus.

Initially, the curve is quite shallow, since the car is slow to start.

During that first second, the distance it travels doesn't change that much.

For the next few seconds, as the car speeds up,

the distance traveled in a given second gets larger,

which corresponds to a steeper slope in this graph.

Then towards the end, when it slows down, that curve shallows out again.

If we were to plot the car's velocity in meters per second as a function of time,

it might look like this bump.

At early times, the velocity is very small.

Up to the middle of the journey, the car builds up to some maximum velocity,

covering a relatively large distance each second.

Then it slows back down towards a speed of zero.

These two curves are definitely related to each other.

If you change the specific distance vs.

time function, you'll have some different velocity vs.

time function.

What we want to understand is the specifics of that relationship.

Exactly how does velocity depend on a distance vs.

time function?

To do that, it's worth taking a moment to think

critically about what exactly velocity means here.

Intuitively, we all might know what velocity at a given moment means,

it's just whatever the car's speedometer shows in that moment.

Intuitively, it might make sense that the car's velocity should be higher at times when

this distance function is steeper, when the car traverses more distance per unit time.

But the funny thing is, velocity at a single moment makes no sense.

If I show you a picture of a car, just a snapshot in an instant,

and I ask you how fast it's going, you'd have no way of telling me.

What you'd need are two separate points in time to compare.

That way you can compute whatever the change in distance across those times is,

divided by the change in time.

Right?

I mean, that's what velocity is, it's the distance traveled per unit time.

So how is it that we're looking at a function for velocity that

only takes in a single value of t, a single snapshot in time?

It's weird, isn't it?

We want to associate individual points in time with a velocity,

but actually computing velocity requires comparing two separate points in time.

If that feels strange and paradoxical, good!

You're grappling with the same conflicts that the fathers of calculus did.

And if you want a deep understanding for rates of change, not just for a moving car,

but for all sorts of things in science, you're going to need to resolve this apparent

paradox.

First, I think it's best to talk about the real world,

and then we'll go into a purely mathematical one.

Let's think about what the car's speedometer is probably doing.

At some point, say 3 seconds into the journey,

the speedometer might measure how far the car goes in a very small amount of time,

maybe the distance traveled between 3 seconds and 3.01 seconds.

Then it could compute the speed in meters per second as that tiny

distance traversed in meters divided by that tiny time, 0.01 seconds.

That is, a physical car just side-steps the paradox and

doesn't actually compute speed at a single point in time.

It computes speed during a very small amount of time.

So let's call that difference in time dt, which you might think of as 0.01 seconds,

and let's call that resulting difference in distance ds.

So the velocity at some point in time is ds divided by dt,

the tiny change in distance over the tiny change in time.

Graphically, you can imagine zooming in on some point of this distance vs.

time graph above t equals 3.

That dt is a small step to the right, since time is on the horizontal axis,

and that ds is the resulting change in the height of the graph,

since the vertical axis represents the distance traveled.

So ds divided by dt is something you can think of as the rise

over run slope between two very close points on this graph.

Of course, there's nothing special about the value t equals 3.

We could apply this to any other point in time,

so we consider this expression ds over dt to be a function of t,

something where I can give you a time t and you can give me back the value of this

ratio at that time, the velocity as a function of time.

For example, when I had the computer draw this bump curve here,

the one representing the velocity function, here's what I had the computer actually do.

First, I chose a small value for dt, I think in this case it was 0.01.

Then I had the computer look at a whole bunch of times t between 0 and 10,

and compute the distance function s at t plus dt,

and then subtract off the value of that function at t.

In other words, that's the difference in the distance traveled between the given time,

t, and the time 0.01 seconds after that.

Then you can just divide that difference by the change in time, dt,

and that gives you velocity in meters per second around each point in time.

So with a formula like this, you could give the computer any curve representing any

distance function s of t, and it could figure out the curve representing velocity.

Now would be a good time to pause, reflect, and make sure this idea

of relating distance to velocity by looking at tiny changes makes sense,

because we're going to tackle the paradox of the derivative head on.

This idea of ds over dt, a tiny change in the value of the function s divided by

the tiny change in the input that caused it, that's almost what a derivative is.

And even though a car's speedometer will actually look at a concrete change in time,

like 0.01 seconds, and even though the drawing program here is looking at an actual

concrete change in time, in pure math the derivative is not this ratio ds over dt for a

specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt

approaches 0.

Luckily there is a really nice visual understanding for what it means to ask what

this ratio approaches, Remember, for any specific choice of dt,

this ratio ds over dt is the slope of a line passing through two separate points

on the graph, right?

Well as dt approaches 0, and as those two points approach each other,

the slope of the line approaches the slope of a line that's

tangent to the graph at whatever point t we're looking at.

So the true honest-to-goodness pure math derivative is not the

rise over run slope between two nearby points on the graph,

it's equal to the slope of a line tangent to the graph at a single point.

Now notice what I'm not saying, I'm not saying that the derivative is

whatever happens when dt is infinitely small, whatever that would mean.

Nor am I saying that you plug in 0 for dt.

This dt is always a finitely small non-zero value, it's just that it approaches 0 is all.

I think that's really clever.

Even though change in an instant makes no sense,

this idea of letting dt approach 0 is a really sneaky backdoor

way to talk reasonably about the rate of change at a single point in time.

Isn't that neat?

It's kind of flirting with the paradox of change in

an instant without ever needing to actually touch it.

And it comes with such a nice visual intuition too,

as the slope of a tangent line to a single point on the graph.

And because change in an instant still makes no sense,

I think it's healthiest for you to think of this slope not as some instantaneous

rate of change, but instead as the best constant approximation for a rate of

change around a point.

By the way, it's worth saying a couple words on notation here.

Throughout this video I've been using dt to refer to a tiny change in t with

some actual size, and ds to refer to the resulting change in s,

which again has an actual size, and this is because that's how I want you to

think about them.

But the convention in calculus is that whenever you're using the letter d like this,

you're kind of announcing your intention that eventually you're

going to see what happens as dt approaches 0.

For example, the honest-to-goodness pure math derivative is written as ds divided by dt,

even though it's technically not a fraction per se,

but whatever that fraction approaches for smaller and smaller nudges in t.

I think a specific example should help here.

You might think that asking about what this ratio approaches

for smaller and smaller values would make it much more difficult to compute,

but weirdly it kind of makes things easier.

Let's say you have a given distance vs time function that happens to be exactly t cubed.

So after 1 second the car has traveled 1 cubed equals 1 meters,

after 2 seconds it's traveled 2 cubed, or 8 meters, and so on.

Now what I'm about to do might seem somewhat complicated,

but once the dust settles it really is simpler,

and more importantly it's the kind of thing you only ever have to do once in calculus.

Let's say you wanted to compute the velocity, ds divided by dt,

at some specific time, like t equals 2.

For right now let's think of dt as having an actual size,

some concrete nudge, we'll let it go to 0 in just a bit.

The tiny change in distance between 2 seconds and 2 plus dt

seconds is s of 2 plus dt minus s of 2, and we divide that by dt.

Since our function is t cubed, that numerator looks like 2 plus dt cubed minus 2 cubed.

And this is something we can work out algebraically.

Again, bear with me, there's a reason I'm showing you the details here.

When you expand that top, what you get is 2 cubed plus 3 times 2 squared dt

plus 3 times 2 times dt squared plus dt cubed, and all of that is minus 2 cubed.

Now there's a lot of terms, and I want you to remember that it looks like a mess,

but it does simplify.

Those 2 cubed terms cancel out.

Everything remaining here has a dt in it, and since there's a dt on the bottom there,

many of those cancel out as well.

What this means is that the ratio ds divided by dt has boiled down into

3 times 2 squared plus 2 different terms that each have a dt in them.

So if we ask what happens as dt approaches 0, representing the idea of looking at a

smaller and smaller change in time, we can just completely ignore those other terms.

By eliminating the need to think about a specific dt,

we've eliminated a lot of the complication in the full expression.

So what we're left with is this nice clean 3 times 2 squared.

You can think of that as meaning that the slope of a line tangent to

the point at t equals 2 of this graph is exactly 3 times 2 squared, or 12.

And of course, there's nothing special about the time t equals 2.

We could more generally say that the derivative

of t cubed as a function of t is 3 times t squared.

Now take a step back, because that's beautiful.

The derivative is this crazy complicated idea.

We've got tiny changes in distance over tiny changes in time,

but instead of looking at any specific one of those,

we're talking about what that thing approaches.

I mean, that's a lot to think about.

And yet what we've come out with is such a simple expression, 3 times t squared.

And in practice, you wouldn't go through all this algebra each time.

Knowing that the derivative of t cubed is 3t squared is one of those things that all

calculus students learn how to do immediately without having to re-derive it each time.

And in the next video, I'm going to show you a nice way to think about

this and a couple other derivative formulas in really nice geometric ways.

But the point I want to make by showing you all of the algebraic guts

here is that when you consider the tiny change in distance caused by a

tiny change in time for some specific value of dt, you'd have kind of a mess.

But when you consider what that ratio approaches as dt approaches 0,

it lets you ignore much of that mess, and it really does simplify the problem.

That right there is kind of the heart of why calculus becomes useful.

Another reason to show you a concrete derivative like this is that it

sets the stage for an example of the kind of paradoxes that come about

if you believe too much in the illusion of instantaneous rate of change.

So think about the actual car traveling according to this t cubed distance function,

and consider its motion at the moment t equals 0, right at the start.

Now ask yourself whether or not the car is moving at that time.

On the one hand, we can compute its speed at that point using the derivative,

3t squared, which for time t equals 0 works out to be 0.

Visually, this means that the tangent line to the graph at that point is perfectly flat,

so the car's quote-unquote instantaneous velocity is 0,

and that suggests that obviously it's not moving.

But on the other hand, if it doesn't start moving at time 0, when does it start moving?

Really, pause and ponder that for a moment.

Is the car moving at time t equals 0?

Do you see the paradox?

The issue is that the question makes no sense.

It references the idea of change in a moment, but that doesn't actually exist.

That's just not what the derivative measures.

What it means for the derivative of a distance function to be 0 is that the best

constant approximation for the car's velocity around that point is 0 m per second.

For example, if you look at an actual change in time,

say between time 0 and 0.1 seconds, the car does move.

It moves 0.001 m.

That's very small, and importantly, it's very small compared to the change in time,

giving an average speed of only 0.01 m per second.

And remember, what it means for the derivative of this motion to be 0 is that

for smaller and smaller nudges in time, this ratio of m per second approaches 0.

But that's not to say that the car is static.

Approximating its movement with a constant velocity of 0 is,

after all, just an approximation.

So whenever you hear people refer to the derivative as an instantaneous rate of change,

a phrase which is intrinsically oxymoronic, I want you to think of that as a

conceptual shorthand for the best constant approximation for rate of change.

In the next couple videos, I'll be talking more about the derivative,

what it looks like in different contexts, how do you actually compute it,

why is it useful, things like that, focusing on visual intuition as always.