11. More Calculus with Python

Overview

In this chapter, you will learn how to calculate the length of a curve, given its equation. You'll be introduced to partial derivatives in three dimensions and how to use them to calculate the area of a surface. Following in the footsteps of the mathematicians of the Middle Ages, you'll use an infinite series to calculate constants such as pi and determine the interval of convergence of a series. Like the mathematicians and machine learning engineers of the modern day, you'll learn how to find the minimum point on a surface using partial derivatives. By the end of this chapter, you will be able to use calculus to solve a variety of mathematical problems.

Introduction

In the previous chapter, we learned how to calculate derivatives and integrals. Now, we're going to use those tools to find the lengths of curves and spirals and extend this reasoning to three dimensions to find the area of a complicated surface. We'll also look at a common tool that's used in calculus, the infinite series, which is used to calculate important constants and approximate complicated functions. Finally, we'll look at an important idea in machine learning: finding the minimum point on a curve. When you use a neural network, you create a kind of "error function" and work hard to find the point on the surface that gives the minimum error. We'll create our own kind of gradient descent function to keep traveling downward until we're at the bottom of the surface.

Length of a Curve

A major use of derivatives and integrals is finding the length of a curve. There's a formula for this:

Figure 11.1: Formula to calculate the length of a curve

The preceding formula contains an integral and a derivative. To find the length of a curve, we'll need both our derivative and integral functions. Copy and paste them into your code if you don't have them yet:

from math import sqrt
 
def derivative(f,x):
    """Returns the value of the derivative of     the function at a given x-value."""
    delta_x = 1/1000000
    return (f(x+delta_x) - f(x))/delta_x
 
def trap_integral(f,a,b,num):
    """Returns the sum of num trapezoids     under f between a and b"""
    width = (b-a)/num
    area = 0.5*width*(f(a) + f(b) + 2*sum([f(a+width*n) \      ...

Length of a Spiral

What about spirals, which are expressed in polar coordinates, where r, the distance from the origin, is a function of the theta (θ) angle that's made with the x axis? We can't use our x and y functions to measure the spiral shown in the following diagram:

Figure 11.10: An Archimedean spiral

What we have in the preceding diagram is a spiral that starts at (5,0) and makes 7.5 turns, ending at (11,π). The formula for that curve is r(θ) = 5 + 0.12892θ. The number of radians turned is 7.5 times 2π, which is 15π. We're going to use the same idea as in the previous section: we're going to find the length of the straight line from r(θ) to r(θ+step) for some tiny step in the central angle, as shown in the following diagram:

Figure 11.11: Approximating the length of a tiny part of the curve

The opposite side to the central angle of the triangle shown in the...

Area of a Surface

Let's learn how to take this from two to three dimensions and calculate the area of a 3D surface. In Chapter 10, Foundational Calculus with Python, we learned how to calculate the area of a surface of revolution, but this is a surface where the third dimension, z, is a function of the values of x and y.

The Formulas

The traditional, algebraic way to solve this analytically is given by a double integral over a surface:

Figure 11.18: Formula to calculate area of a surface

Here, z = f(x,y) or (x,y,f(x,y)). Those curly d's are deltas, meaning we'll be dealing with partial derivatives. Partial derivatives are derivatives but only with respect to one variable, even if the function is dependent on more than one variable. Here's a function that returns the partial derivative of a function, f, with respect to a variable, u, at a specific point (v,w). Depending on which variable we're interested in, x or y, the function...

Infinite Series

Mathematicians have often run into functions that are too complicated for them to solve or otherwise deal with, and approximations have always been an important component in doing math. For mathematicians trying to take derivatives and integrals algebraically, many expressions have no nice neat solutions, derivatives, integrals, and so on. In general, no differential equations that scientists come across in real life have algebraic solutions, so they have to use other methods. More on differential equations later, but there's an important family of approximations that use easy functions to approximate hard ones.

Polynomial Functions

It's easy to solve, differentiate, and integrate polynomial equations—ones such as y = x2 and even the following equation:

Figure 11.28: A polynomial equation

The terms are all added (or subtracted) one after the other, and there are no trigonometric, logarithmic, or exponential functions involved...

Summary

In the previous chapter, we learned the power of derivatives and integrals, so in this chapter, we built on those tools to solve some pretty difficult problems, such as the length of a spiral and the area of a 3D surface. We even extended derivatives and integrals to three dimensions by introducing partial derivatives. In a calculus class, we would be using lots of algebra in order to use these tools, but by using Python, we modeled the situation and tested our functions. We created variables that will contain our changing values and looped through the calculations millions of times, if necessary. To the mathematicians of previous centuries, this would have seemed like some kind of magic lamp.

In the next chapter, we'll deal with more changing rates and amounts and avoid a lot of algebra by using Python. We'll find out how much salt is in an ever-changing mixture, when and where a predator will catch its prey, and how long we'll have to invest our money to...