Constructing a continuous function

Exponential Functions

Introduction

Recall that previously we studied functions of the form a2+bx+c and mx+b. These are examples of polynomials. Polynomials are said to be “linear combinations” of monomials.

A linear combination is just the sum of constant multiples of simpler functions. For example, ax2+bx+c is a linear combination of x2, x, and 1.

Each of these terms (ax2, bx, and c) are said to be monomials and have the form axn where a and n are constant as x varies. Some concrete examples would be 2, 3x2, 5x3, 7x21, etc.. Another variety of functions involving exponents are the Exponential functions. They have the form abx, where a and b are constant, as x varies.

Notice that monomials have a variable base, whereas exponentials have a variable exponent. For example, x3 is a monomial whereas 3x is an exponential.

An exponential function's rate of growth (or decay, if the base is less than 1) varies directly with it's current value. For example, a city of 2 million people should have twice as many babies born this year as a city of 1 million people if thier populations grow exponentially over time.

Figure 1
Exponential functions grow much more quickly than polynomials do. This is because an exponential function's proportional growth rate remains constant, while a polynomial's will decline and barely be greater than 1 for large values of x. For example 3x+1/3x will always equal 3 whereas (x + 1)3/x3 = (1 + 1/x)3 will approach 1 for large values of x. Try to make a table to see for yourself that this is what happens.

We will learn how to analyze exponential functions shortly, but first we will discuss some of their applications.

Financial Appliations

Suppose one invested a principle amount (P) for compounded annually at 12 percent interest.
After 1 year (after adding interest) it would be worth P+0.12P = P(1+0.12).
After 2 years, it would be worth
P(1+0.12)+0.12(P(1+0.12)) = (1+0.12)(P(1+0.12)) = P(1+0.12)2.
After 3 years P(1+0.12)3, and so forth.

It appears that compound interest can be modeled as follows: PN = P(1+I)N, where a principle (P) is invested at an interest rate (I) for a number of compunding periods (N) yeilding a future value (PN).

Quite often though, one doesn't invest a lump sum at the start of an investment period, rather one invests the same amount of money per month or per year. Think about paying off a car loan, or investing in a retirement or education savings plan. This type of investment is called an annuity and can be modeled as a geometric series. A geometric series has the form: ar+ar2+ar3+...+arn. Later, We will be discussing how to calculate the sum of a geometric series and how it applies to annuities.

Growth and decay

Exponential functions can be use to model population growth. For example, predicting the world's population in 2020 or determining the concentration of bacteria in a sample after 60 minutes are both examples of exponential growth. They growth pattern is exponential because the rate of growth depends on the current population.

Another use for exponential functions is for modeling exponential decay. A hot frying pan will cool quickly in the surrounding air, but as it cools, its rate of cooling declines as its temerature approaches the room temperature. The temperature difference is an example of exponential decay. In carbon-14 dating, the concentation of carbon-14 isotopes are compared to that of the more common carbon-12. By comparing this ratio to the expected half-life of carbon-14, scientists can determine a fossil's age.

Sometimes, a population appears to grow exponentially, but reason would dictate that this rate cannot be maintained. For example, bacteria can grow in a sample exponentially until its food supply begins to run out, then its rate of growth will slow. This type of growth is called logistical growth. We will discuss expontential growth and decay later, but we will hold off on logisitical growth until the section on grade 12 calculus as it involves differential equations.

Statistics

Exponential functions are used to model many probability distributions in the field of Statistics. One of the most important is called the Normal Distribution (also refered to as the Gaussian Distribution or the bell shaped curve). Many quanitities in the real world are normally distributed. Examples include human height, weight, and IQ scores. It can also be shown that given a sample from ANY population (regardless of its own probability distribution) that the means (or averages) of the samples become more normally distributed as the size of the samples increase.

The standard normal distribution (where the mean is 0.0 and the standard deviation 1.0) can be modeled as:

We will derive this formula in the grade 12 section of this website.

The Catenary

We won't be considering the catenary past this introduction, but it is included here just for general interest. The shape of a wire suspended by 2 poles influenced only by its own weight is called a catenary. Not surprizingly, some suspension bridges settle into the shape of a catenary. Mathematically, a catenary can be modeled by exponential functions and has the form
This is an example of a Hyperbolic Function. Hyperbolic Functions will be discussed when we cover grade 12 calculus.
Figure 2
These images of suspended telephone lines in the Welsh countryside, the dome of the U.S. Capitol Building, the Gateway Arch in St. Louis, an igloo, and the Capilano Suspension Bridge in North Vancouver are all examples of catenaries. The stacked lego blocks also follow the approximate shape of a catenary. Notice that the blue block at the top is actually past the edge of the table. A bubble formed between 2 rings of equal radius will have a catenary in cross-section.

A catenary is also the ideal shape for a freestanding arch or dome of constant thickness. Unlike a dome with a semi-circular cross section, a catenary dome's sides will not bulge out, nor will the top cave in. In other words, it is a shape that will support itself.

A soap film bubble that forms between 2 rings of equal radius will form the shape of a catenoid (a 3D shape with a catenary in cross section). A catenoid is the shape that will minimize the surface area. Other minimal surfaces include a sphere and a plane. These correspond to freely floating bubbles and the film that is initially inside the ring of a bubble wand.

Finally a word on the relationship between quadratic functions and catenaries. If you roll a parabola on a straight line, the path traced out by the parabola's focus will form a catenary. Also noteworth is the fact that a formula for a circle also involves an x2 term (x2+y2=r2). If there is some relationship between quadratics and circles and also between quadratics and exponentials, then one may suppose that there is some relationship between exponentials and circles. There is in fact a relationship:

eix = cos x + isin x, where i2 = -1.

This formula was first published in 1748 by the great mathemetician Leonhard Euler and is one of the most important formulas ever discovered. This formula will make a lot more sense when we cover Imaginary and Complex numbers. The irrational constant e is approximately 2.7182818..., and will be discussed breifly at the end of the section on exponentials.

Constructing a continuous function