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 + a2x2 + ... + 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 + a2x2 + a3x3 + ... + anxn
f1n(x)= a1 + 2a2x + 3a3x2 +... + 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.
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