Higher order derivatives | Chapter 10, Essence of calculus

Summary

This educational video script delves into the concept of higher order derivatives, particularly focusing on second derivatives. It explains how the second derivative represents the rate of change of the slope, indicated by the curvature of a graph. Positive values suggest an increasing slope, while negative values indicate a decreasing one. The script also touches upon the third derivative, humorously termed 'jerk,' which measures changes in acceleration. The discussion is set to segue into the application of higher order derivatives in approximating functions, specifically through Taylor series.

Takeaways

• 📕 The script introduces the concept of higher order derivatives, specifically focusing on the second derivative as a precursor to discussing Taylor series.
• 💵 The first derivative of a function represents the slope of the graph at any given point, indicating the rate of change.
• 📗 The second derivative is the derivative of the first derivative, indicating how the slope of the graph is changing.
• 📐 A positive second derivative indicates the graph is curving upwards, meaning the slope is increasing, while a negative second derivative indicates the graph is curving downwards, meaning the slope is decreasing.
• 💲 The notation for the second derivative is often written as \( \frac{d^2f}{dx^2} \), which represents the limit of the ratio of the change in the function's change to the change in \( x \) as \( x \) approaches zero.
• 📊 The script uses the analogy of movement along a line to explain the second derivative as acceleration, where the first derivative represents velocity.
• 💵 The third derivative, humorously referred to as 'jerk', indicates how the acceleration is changing, showing whether the rate of speed increase or decrease is itself increasing or decreasing.
• 📖 Higher order derivatives are essential for approximating functions, which is crucial for understanding the upcoming topic of Taylor series.
• 💲 The script provides a visual example to help understand the concept of the second derivative, comparing graphs with different slopes and curvatures.
• 💵 The second derivative's sign (positive or negative) corresponds to the sensation of speeding up or slowing down in motion, respectively.
• 📗 The concept of 'jerk' is introduced as the third derivative, which is significant for understanding changes in acceleration.

Q & A

What is the main focus of the next chapter after discussing Taylor series?

The main focus of the next chapter is higher order derivatives, particularly the second derivative, and its application in the context of graphs and motion.

What is the significance of higher order derivatives in the context of the script?

Higher order derivatives are significant as they provide insights into how the slope of a function is changing, which is crucial for understanding concepts like acceleration and jerk in physics.

How is the second derivative related to the curvature of a graph?

The second derivative is related to the curvature of a graph by indicating whether the graph is curving upwards (positive second derivative) or downwards (negative second derivative).

What does a positive second derivative at a certain point on a graph indicate?

A positive second derivative at a certain point on a graph indicates that the slope is increasing at that point, suggesting the graph is curving upwards.

What is the physical interpretation of the second derivative when considering motion?

The second derivative has a physical interpretation as acceleration when considering motion, as it tells you the rate at which the velocity is changing.

What is the term used for the third derivative in the context of motion?

The third derivative in the context of motion is humorously referred to as 'jerk', which indicates how the acceleration is changing.

How does the second derivative help in approximating functions?

The second derivative helps in approximating functions by providing information about the concavity and rate of change of the function, which is useful in series expansion techniques like Taylor series.

What is the standard notation for the second derivative?

The standard notation for the second derivative is written as \( \frac{d^2f}{dx^2} \), which represents the derivative of the derivative with respect to \( x \).

What does the term 'ddf' in the script refer to?

The term 'ddf' refers to the change in the change of the function, which is proportional to the square of the small change in \( x \) (dx^2), and it is used to define the second derivative.

How does the script suggest visualizing the process of calculating the second derivative?

The script suggests visualizing the process by taking two small steps to the right with a size of dx, observing the change in the function (df1 and df2), and then calculating the difference between these changes (ddf).

What is the practical application of higher order derivatives mentioned in the script?

The practical application of higher order derivatives mentioned in the script is their use in approximating functions, which is particularly relevant to the upcoming discussion on Taylor series.

Outlines

00:00

📈 Understanding Higher Order Derivatives

This paragraph introduces the concept of higher order derivatives, specifically focusing on the second derivative. It explains that the second derivative is the derivative of the first derivative, indicating how the slope of a function's graph is changing. The paragraph uses the analogy of a graph's curvature to illustrate the concept: a positive second derivative corresponds to an upward curve where the slope is increasing, while a negative second derivative corresponds to a downward curve where the slope is decreasing. The notation for the second derivative is explained, with the standard notation being d²f/dx². The paragraph also touches on the physical interpretation of the second derivative as acceleration in the context of motion, and the third derivative as 'jerk', which indicates changes in the acceleration.

05:06

🔍 Approximating Functions with Higher Order Derivatives

The second paragraph discusses the application of higher order derivatives in approximating functions, which is a key topic in the upcoming chapter on Taylor series. It suggests that the positive second derivative in the first half of a journey indicates acceleration, providing a physical sensation of being pushed back into a car seat. Conversely, a negative second derivative indicates deceleration or negative acceleration. The paragraph concludes by emphasizing the utility of higher order derivatives in function approximation, hinting at the importance of this concept in the context of the Taylor series, which will be explored in more detail in the subsequent chapter.

Keywords

💡Taylor series

A Taylor series is a mathematical representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In the context of the video, Taylor series is mentioned as the next chapter's topic, suggesting that higher-order derivatives will be instrumental in approximating functions, which is the essence of the series.

💡Higher-order derivatives

Higher-order derivatives refer to the successive differentiation of a function beyond the first derivative. The video script introduces these as essential for understanding changes in the slope of a function's graph, with the second derivative indicating how the slope itself changes. This concept is crucial for comprehending the curvature and behavior of functions.

💡Second derivative

The second derivative of a function is the derivative of its first derivative. It measures how the slope of the function's graph is changing. In the video, the second derivative is used to describe the curvature of a graph, with a positive value indicating upward curvature and a negative value indicating downward curvature.

💡Slope

Slope, in the context of calculus, is the steepness of a line or curve at a particular point. The video explains that the first derivative of a function represents the slope of its graph at any given point, with a steep slope corresponding to a high derivative value.

💡Curvature

Curvature, in the context of the video, refers to the bending or deviation of a curve from being straight. It is directly related to the second derivative, where positive curvature indicates the graph is bending upwards, and negative curvature indicates it is bending downwards.

💡Notation

Notation in calculus is a system of symbols used to represent mathematical operations and concepts. The video discusses the notation for the second derivative, which is written as 'd²f/dx²', indicating the derivative of the derivative with respect to 'x'. This notation is essential for understanding the operations being performed on the function.

💡dx

In calculus, 'dx' represents an infinitesimally small change in the independent variable 'x'. The video uses 'dx' to describe the process of taking small steps in 'x' to calculate the change in the function and its derivative, which is foundational to the concept of derivatives.

💡Velocity

Velocity is the rate of change of position with respect to time. The video uses velocity as an analogy to explain the first derivative, showing how it can be represented graphically as the slope of a distance-time graph.

💡Acceleration

Acceleration is the rate at which velocity changes over time. The video explains that the second derivative, in the context of motion, represents acceleration, which is the change in velocity. This is a key concept in physics and is used to illustrate the practical application of the second derivative.

💡Jerk

Jerk, in physics, is the rate of change of acceleration with respect to time. The video humorously introduces the third derivative as 'jerk', which is a measure of how quickly acceleration changes. This term is used to further explain the concept of higher-order derivatives and their applications.

💡Approximation

Approximation in mathematics refers to the process of finding a value that is close to a given value or quantity. The video mentions that higher-order derivatives are useful for approximating functions, which ties into the broader theme of the series and the upcoming discussion on Taylor series.

Highlights

Introduction to higher order derivatives for those familiar with second derivatives.

Explanation of higher order derivatives for those who haven't encountered them yet.

Derivative as the slope of a graph at a point.

Second derivative as the rate of change of the slope.

Second derivative's relation to the curvature of the graph of a function.

Positive second derivative indicates an increasing slope.

Negative second derivative indicates a decreasing slope.

Zero second derivative at points with no curvature.

Standard notation for the second derivative: d²f/dx².

Intuitive understanding of second derivative through small steps in function values.

Second derivative as the rate of change of velocity, or acceleration.

Third derivative, also known as jerk, represents the rate of change of acceleration.

Higher order derivatives help in approximating functions, especially in Taylor series.

Positive second derivative indicates speeding up, like being pushed back into a car seat.

Negative second derivative indicates slowing down, or negative acceleration.

Jerk as a measure of how the strength of acceleration is changing.

The utility of higher order derivatives in understanding motion and approximating functions.

Transcripts

In the next chapter about Taylor series, I make

frequent reference to higher order derivatives.

And if you're already comfortable with second derivatives,

third derivatives, and so on, great!

Feel free to just skip ahead to the main event now.

You won't hurt my feelings.

But somehow, I've managed not to bring up higher

order derivatives at all so far in this series.

So for the sake of completeness, I thought I'd give you

this little footnote just to go over them very quickly.

I'll focus mainly on the second derivative, showing what it looks like in the context

of graphs and motion, and leave you to think about the analogies for higher orders.

Given some function f of x, the derivative can be

interpreted as the slope of this graph above some point, right?

A steep slope means a high value for the derivative,

a downward slope means a negative derivative.

So the second derivative, whose notation I'll explain in just a moment,

is the derivative of the derivative, meaning it tells you how that slope is changing.

The way to see that at a glance is to think about how the graph of f of x curves.

At points where it curves upwards, the slope is increasing,

and that means the second derivative is positive.

At points where it's curving downwards, the slope is decreasing,

so the second derivative is negative.

For example, a graph like this one has a very positive second derivative at the point 4,

since the slope is rapidly increasing around that point,

whereas a graph like this one still has a positive second derivative at the same point,

but it's smaller, the slope only increases slowly.

At points where there's not really any curvature, the second derivative is just 0.

As far as notation goes, you could try writing it like this,

indicating some small change to the derivative function,

divided by some small change to x, where as always the use of this letter d

suggests that what you really want to consider is what this ratio approaches as dx,

both dx's in this case, approach 0.

That's pretty awkward and clunky, so the standard is

to abbreviate this as d squared f divided by dx squared.

And even though it's not terribly important for getting an intuition for the second

derivative, I think it might be worth showing you how you can read this notation.

To start off, think of some input to your function,

and then take two small steps to the right, each one with a size of dx.

I'm choosing rather big steps here so we'll be able to see what's going on,

but in principle keep in the back of your mind that dx should be rather tiny.

The first step causes some change to the function, which I'll call df1,

and the second step causes some similar but possibly slightly different change,

which I'll call df2.

The difference between these changes, the change in how the function changes,

is what we'll call ddf.

You should think of this as really small, typically proportional to the size of dx2.

So if, for example, you substituted in 0.01 for dx,

you would expect this ddf to be about proportional to 0.0001.

The second derivative is the size of this change to the change,

divided by the size of dx2, or more precisely,

whatever that ratio approaches as dx approaches 0.

eleration. Given some movement along a line, suppose you have some function that

records the distance traveled versus time, maybe its graph looks like this,

steadily increasing over time. Then its derivative tells you velocity at each

point in time, for example the graph might look like this bump,

increasing up to some maximum, and decreasing back to zero.

So the second derivative tells you the rate of

Maybe the most visceral understanding of the second

derivative is that it represents acceleration.

Given some movement along a line, suppose you have some function

that records the distance traveled versus time,

maybe its graph looks something like this, steadily increasing over time.

Then its derivative tells you velocity at each point in time,

for example the graph might look like this bump, increasing up to some maximum,

and decreasing back to zero.

The third derivative, and this is not a joke, is called jerk. So if the jerk is not zero,

it means that the strength of the acceleration itself is changing.

One of the most useful things about higher order derivatives is how they help us in

approximating functions,

In this example, the second derivative is positive for the first half of the journey,

which indicates speeding up, that's the sensation of being pushed back into

your car seat, or rather, having the car seat push you forward.

A negative second derivative indicates slowing down, negative acceleration.

The third derivative, and this is not a joke, is called jerk.

So if the jerk is not zero, it means the strength of the acceleration itself is changing.

One of the most useful things about higher order derivatives is

how they help us in approximating functions, which is exactly the

topic of the next chapter on Taylor series, so I'll see you there.