Last updated June 19, 2021

Measurement of the rate of change is an inseparable concept in differential calculus, which deals with the mathematics of change and infinity. It allows us to find the relationship between two variables and how they affect each other.

Measurement of the rate of change is also essential for machine learning, such as the application of gradient descent as an optimization algorithm to train the neural network model.

In this tutorial, you will discover the rate of change as one of the key concepts in computation and the importance of measuring it.

After completing this tutorial, you will know:

  • How to measure the rate of change of linear and nonlinear functions.
  • Why measuring the rate of change is an important concept in different fields.

Let’s get started.

Key concepts in computation: rate of change
Picture: Spencer Everett, some rights reserved.

Curriculum Overview

This tutorial is divided into two parts; they are:

  • The pace of change
  • The importance of measuring the rate of change

The pace of change

The rate of change determines one variable in relation to another.

Consider a moving object that moves twice vertically, marked Y, because it is horizontal, marked x. Mathematically, this can be expressed as follows:

𝛿Y = 2𝛿x

Greek letter delta, 𝛿, is often used to denote differential or change. Thus, the above equation defines the relationship between the change xstrain in relation to change Ylocation of the moving object.

This change x and Yaddresses can be drawn with a straight line xY coordinate system.

Linear function line diagram

In this graphical representation of the motion of an object, it represents the rate of change gradient or its slope. Because the line can be seen rise 2 units for each individual unit passing to the right, its rate of change or slope is equal to 2.

Prices and slopes are simple. The previous velocity examples can be drawn in the xy coordinate system, where each velocity is displayed as an inclination.

Page 38, Calculus Essentials for Dummies, 2019.

Putting it all together, we see that:

rate of change = 𝛿Y / 𝛿x = rise / run = slope

If we have to look at two specific points, P1 = (2, 4) and P2 = (8, 16), with this line we can confirm that the slope is equal to:

slope = 𝛿Y / 𝛿x = (Y2Y1) / (x2x1) = (16-4) / (8-2) = 2

In this particular example, the rate of change represented by the slope is positive because the direction of the line increases to the right. However, the rate of change can also be negative if the direction of the line decreases, which means that the value Y would decrease the value x grow. In addition, when the value Y remains constant x grow, we would say we have zero the pace of change. If otherwise the value x remains constant Y grow, we consider the area of ​​change to be infinite, because the slope of the vertical line is considered undefined.

So far, we have considered the simplest example of obtaining a direct and thus linear function with a constant slope. However, not all operations are that simple, and if they were, no calculation would be required.

Calculation is the math of change, so now is a good time to move on to parabolas, curves with changing slopes.

Page 39, Calculus Essentials for Dummies, 2019.

Consider a simple nonlinear function, parabolism:

Y = (1/4) x2

In contrast to the constant slope that characterizes a straight line, we can see how this parabola becomes steeper as we move to the right.

Parabolan plot
Taken from Calculus Essentials for Dummies

Let us remember that the computational method allows us to analyze a curved shape by cutting it into many infinitely straight pieces arranged side by side. If we have to consider one of these songs at some point, P, with the curved shape of the parabola, we see that we find ourselves recalculating the rate of change as a straight line. It is important to keep in mind that the rate of change in a parabola depends on a certain point P, which we happened to consider first.

For example, if we have to take into account a straight line passing through a point, P = (2, 1), we find that the rate of change at this point in the parabola is:

rate of change = 𝛿Y / 𝛿x = 1/1 = 1

If we were to consider a different point in the same parable, at P = (6, 9), we find that the rate of change at this stage is equal to:

rate of change = 𝛿Y / 𝛿x = 3/1 = 3

A straight line touch curve at a certain point, P, known as tangent line, while the process of calculating the rate of change of a function is also known as finding it derivative.

A derivative is simply a measure of how much one thing changes compared to another – and that is the interest rate.

Page 37, Calculus Essentials for Dummies, 2019.

Although we have considered a simple parabola in this example, we can similarly use a calculator to analyze more complex nonlinear functions. The concept of calculating the instantaneous rate of change at different tangential points on the curve remains the same.

We will meet one such example when we come to train the neural network with a gradient descent algorithm. As an optimization algorithm, the gradient descent will iteratively lower the error function toward its global minimum, updating the neural network weights each time to better model exercise data. The error function is typically nonlinear and may include many local minima and saddle points. To find its way downhill, the gradient descent algorithm calculates the instantaneous slope error at different points until it reaches the point where the error is smallest and the rate of change is zero.

The importance of measuring the rate of change

So far, we have taken into account the rate of change per unit xY coordinate system.

But the interest rate can be anything to anything.

Page 38, Calculus Essentials for Dummies, 2019.

For example, in connection with neural network training, we have seen that the error gradient is calculated as the error change with respect to a given weight of neural network.

There are many different areas where measuring the rate of change is also an important concept. A few examples are:

  • In physics speed is calculated as the change in position per unit time.
  • In signal digitization sampling frequency is calculated as the number of signal samples per second.
  • In the calculation bit rate is the number of bits processed per computer per unit time.
  • In financing exchange refers to the value of one currency relative to another.

In either case, each rate is a derivative, and each derivative is an interest rate.

Page 38, Calculus Essentials for Dummies, 2019.

Further reading

This section provides more resources on the topic if you want to go deeper.

Books

Summary

In this tutorial, you will find the rate of change as one of the key concepts in computation and the importance of measuring it.

In particular, you will learn:

  • Measurement of the rate of change is an inseparable concept in differential calculus that allows us to find one variable with respect to another.
  • This is an important concept that can be applied in many fields, one of which is machine learning.

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