This essay Gods Gift to Calculators: The Taylor Series has a total of 569 words and 4 pages.
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
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
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.
Topics Related to Gods Gift to Calculators: The Taylor Series
Exponentials, Functions and mappings, Derivative, Differential calculus, Mathematical analysis, Rates, Polynomial, Taylor series, Taylors theorem, Exponential function, Exponentiation, Approximation theory
Essays Related to Gods Gift to Calculators: The Taylor Series
Isiah M. WarnerIsiah M. Warner Isiah M. Warner was born in DeQuincy, Louisiana on July 20, 1946. At the age of two, he moved to Bunkie, Louisiana and attended Carver Elementary. His interest in science developed when he orally sampled kerosene to determine why it had an odd smell and was able to produce light. At the age of twelve, he received his first chemistry set. Upon entering Carver high school, Isiah enrolled in every science class available. He graduated valedictorian in June of 1964. During the summer