Gods Gift To Calculators: The Taylor Series Essay, Research Paper

Gods Gift to Calculators: The Taylor Series

It is incredible how far calculators have come since my parents were in

college, which was when the square root key came out. Calculators since then

have evolved into machines that can take natural logarithms, sines, cosines,

arcsines, and so on. The funny thing is that calculators have not gotten any

“smarter” since then. In fact, calculators are still basically limited to the

four basic operations: addition, subtraction, multiplication, and division! So

what is it that allows calculators to evaluate logs, trigonometric functions,

and exponents? This ability is due in large part to the Taylor series, which

has allowed mathematicians (and calculators) to approximate functions,such as

those given above, with polynomials. These polynomials, called Taylor

Polynomials, are easy for a calculator manipulate because the calculator uses

only the four basic arithmetic operators.

So how do mathematicians take a function and turn it into a polynomial

function? Lets find out. First, lets assume that we have a function in the form

y= f(x) that looks like the graph below.

We’ll start out trying to approximate function values near x=0. To do

this we start out using the lowest order polynomial, f0(x)=a0, that passes

through the y-intercept of the graph (0,f(0)). So f(0)=ao.

Next, we see that the graph of f1(x)= a0 + a1x will also pass through x=

0, and will have the same slope as f(x) if we let a0=f1(0).

Now, if we want to get a better polynomial approximation for this

function, which we do of course, we must make a few generalizations. First, we

let the polynomial fn(x)= a0 + a1x + a2×2 + … + anxn approximate f(x) near x=0,

and let this functions first n derivatives match the the derivatives of f(x) at

x=0. So if we want to make the derivatives of fn(x) equal to f(x) at x=0, we

have to chose the coefficients a0 through an properly. How do we do this?

We’ll write down the polynomial and its derivatives as follows.

fn(x)= a0 + a1x + a2×2 + a3×3 + … + anxn

f1n(x)= a1 + 2a2x + 3a3×2 +… + nanxn-1

f2n(x)= 2a2 + 6a3x +… +n(n-1)anxn-2

.

.

f(n)n(x)= (n!)an

Next we will substitute 0 in for x above so that

a0=f(0)a2=f2(0)/2!an=f(n)(0)/n!

Now we have an equation whose first n derivatives match those of f(x) at

x=0.

fn(x)= f(0) + f1(0)x + f2(0)x2/2! + … + f(n)(0)xn/ n!

This equation is called the nth degree Taylor polynomial at x=0.

Furthermore, we can generalize this equation for x=a instead of just

approximating about 0.

fn(x)= f(a) + f1(a)(x-a) + f2(a)(x-a)2/2! + … + f(n)(a)(x-a)n/ n!

So now we know the foundation by which mathematicians are able to design

calculators to evaluate functions like sine and cosine so that we do not have to

rely on a table of values like they did in days past. In addition to the

knowledge of how calculators approximate values of transcendental functions, we

can also see the applications of Taylor series in physics studies. These series

appear in mathematical descriptions of vibrating strings, heat flow,

transmission of electrical current, and motion of a simple pendulum.