12. Intermediate Calculus with Python

Overview

By the end of this chapter, you will be able to solve problems when given equations dealing with the change in a variable. In this chapter, you'll use numerical methods to model populations and temperatures and use differential equations to calculate their past values or predict their future values. You'll learn how to use binary search to guess and check your way to a very accurate solution when you know it's in a specific range of numbers. You'll also model situations where objects move and solve for their future positions when given differential equations for their velocity.

Introduction

Math students often complain that there's no real-world application for the kinds of problems arising in algebra and geometry, such as factoring polynomials or bisecting angles, but the same can't be said for differential equations. Using the tools, you'll learn about in this chapter, you will be able to model and solve real-life problems in physics, electronics, and engineering with differential equations. Python is the perfect tool for mathematicians and scientists who want to be able to crunch numbers and solve problems but don't want to have to get another degree in computer science to do so. Python is one of the most popular programming languages due to its ease of use and lack of unnecessary abstraction.

By the 1600s, mathematicians had modeled the motion of falling objects with mathematical equations and had set their sights on the planets in outer space. Newton modeled their motion and the equations he came up with referred not only to unknown...

Differential Equations

Solving an equation in a math class usually involves an unknown number, x. The equation hides the value but gives you hints as to how to find the value, such as . But to solve a differential equation, you're only given information regarding the derivative of a function, and you're expected to find the function. It could be something as simple as the following:

Figure 12.1: Finding a function with derivative 2

This means find a function whose derivative is 2. This can also be written as follows:

Figure 12.2: Alternative way to represent derivative of the function

By simple integration, we can find functions that work in this equation because we know the function y = 2x has a derivative of 2. In fact, many related functions, such as y = 2x + 1, y = 2x + 2, y = 2x + 3, and so on, all have a derivative of 2. So, we write a general form, that is, y = 2x + C.

Things get more complicated when we don&apos...

Interest Calculations

There's a crucial tool in the study of differential equations that originated in the study of interest calculations in the middle ages. Let's take a look at the following exercise.

Exercise 12.01: Calculating Interest

A savings account pays 2% annual interest. If $3,500 is invested, how much money is in the account after 5 years?

The formula for simple interest is as follows:

Figure 12.4: Formula for simple interest

Here, I is the interest, P is the principal or the original amount invested, r is the rate of interest or growth, and t is the time the amount has been invested for. By this formula, the interest earned on the amount is I = (3500)(0.02)(5) = $350. Follow these steps to complete this exercise:

1. This is a good opportunity to start a program that will take in an initial amount, interest rate, and time and output the interest earned using the preceding formula:

   def amount(p0,rate,t):
      ...

Population Growth

Differential equations are useful for finding a formula for the population of people, animals, and bacteria at a certain time; for example:

Figure 12.7: Differential equation to calculate population at time t

This differential equation means the rate of change of y is proportional to y, or the population grows proportional to its amount. This is the definition of population growth rate: a fraction or percentage of the population. The solution is similar to our interest problems involving continuous compounding:

Figure 12.8: Differential equation to calculate the rate of change

Exercise 12.06: Calculating the Population Growth Rate – Part 1

In the 1980s, the annual population growth rate in Kenya was 4%. At that rate, how long would it take for the population to double? Follow these steps to complete this exercise:

1. No matter what the starting population, we're looking for t, which makes the factor...

Half-Life of Radioactive Materials

Much like population problems, half-life problems concern a population, but one of atoms of radioactive materials where half the atoms change over time into atoms of a different substance. For example, Carbon-14 decays into Nitrogen-14, and it takes about 5,730 years for half the carbon to decay. This makes radiocarbon dating a crucial tool in everything from archaeology to detecting forged artworks.

Exercise 12.08: Measuring Radioactive Decay

Radium-226 has a half-life of 1,600 years. How much of the radium in a given sample will disappear in 800 years?

The differential equation meaning "the rate of decay of a substance is proportional to the amount of the substance" is expressed like this:

Figure 12.10: Differential equation for calculating rate of decay of a substance

The solution is similar to that for our population problems, except that the decay factor is negative, since the amount decreases:

...

Newton's Law of Cooling

Did you ever wonder how the Crime Scene Investigator (CSI) with the latex gloves on police shows can tell the time of death of the victim? Isaac Newton is credited with figuring out that the cooling of substances follows a differential equation:

Figure 12.15: Differential equation for rate of change of temperature

See how this differential equation is slightly different than the ones we've seen before? Instead of the rate of change of the temperature of the substance being proportional to the temperature of the substance, this says "the rate of change of the temperature of a substance is proportional to the difference between the temperature of the substance and the temperature of the environment." So, if a cup of hot coffee is left in a hot room, its temperature is going to change less quickly than if it's left in a very cold room. Similarly, we know the starting temperature of the body of the victim on the...

Mixture Problems

In algebra, there are word problems where you have to figure out how much material you have to add to a mixture to get a certain concentration or amount. In calculus, naturally, the problem has to be harder: for example, the mixture is changing; material is going into the mixture, and material is going out. You have to find out how much mixture or how much of the solvent is present after a specific amount of time. Let's look at the following exercise to better understand this concept.

Exercise 12.12: Solving Mixture Problems – Part 1

A tank contains 82 gallons of brine in which 18 pounds of salt is dissolved. Brine containing 3 pounds of dissolved salt per gallon flows into the tank at the rate of 5 gallons per minute. The mixture, which is kept uniform by stirring, flows out of the tank at a rate of 2 gallons per minute. How much salt is in the tank at the end of 39 minutes?

As you can imagine, this kind of problem leads to some complicated differential...

Euler's Method

In undergraduate math classes, you're taught all these algebraic methods for taking derivatives and integrals and solving differential equations. We didn't mention Laplace transforms, which are even more complicated ways of solving differential equations algebraically. Now, for the dirty secret about differential equations they don't tell you in school, unless you major in engineering: most differential equations you come across in real life have no analytical solution.

The good news is there have been numerical methods for avoiding messy algebra for hundreds of years, and with the invention of computers, these methods have become standard. Even when there is an analytical solution, numerical methods can be almost as accurate for practical purposes as the analytical method and take a fraction of the time to get a solution.

The idea of Euler's method is very simple:

1. Start at the known point.
2. Calculate the derivative at this point...

Summary

Calculus is a very powerful set of tools for modeling real situations, from the transfer of heat to the motion of planets. It has enabled us to calculate the rate of change of a function in an instant and the area under complicated curves (tasks that seemed impossible using only the tools of algebra and geometry). In this chapter, we've been able to deal with the rate of change of a value (the derivative) as a value in itself, and we've calculated some very accurate results using Python loops and functions. Modeling situations that lead to differential equations, such as the paths of projectiles, was what drove the development of the first electronic computers.

Math classes may still emphasize algebraic solutions to equations, even differential equations, but as we've seen in this chapter, using a computer is a straightforward way to model a real-life situation such as a predator pursuing its prey. We changed variables such as the amount of money in an investment...