Gods Gift To Calculators: The Taylor Series :: essays research papers

Gods Gift to Calculators The Taylor Series     It is incredible how far computing devices consume come since my parents were incollege, which was when the square root key came out. Calculators since then gain evolved into machines that can apportion natural logarithms, sines, cosines,arcsines, and so on. The funny thing is that calculators demand not gotten any"smarter" since then. In fact, calculators are still basically hold in to thefour basic operations addition, subtraction, multiplication, and division Sowhat is it that allows calculators to evaluate logs, trigonometric dish ups,and exponents? This ability is due in large part to the Taylor series, whichhas allowed mathematicians (and calculators) to approximate functions,such asthose given above, with polynomials. These polynomials, called TaylorPolynomials, are easy for a calculator manipulate because the calculator usesonly the four basic arithmetic operators.     So how do mathematicians take a function and turn it into a polynomialfunction? Lets rule out. First, lets assume that we father a function in the formy= f(x) that looks like the graph below.     Well start out trying to approximate function values near x=0. To dothis we start out employ the lowest order polynomial, f0(x)=a0, that passesthrough the y-intercept of the graph (0,f(0)). So f(0)=ao.     Next, we see that the graph of f1(x)= a0 + a1x go out also pass through x=0, and will have the same slope as f(x) if we let a0=f1(0).     Now, if we ask to get a better polynomial approximation for thisfunction, which we do of course, we must make a few generalizations. First, welet the polynomial fn(x)= a0 + a1x + a2x2 + ... + anxn approximate f(x) near x=0,and let this functions first n derivatives represent the the derivatives of f(x) atx=0. So if we want to make the derivatives of fn(x) equal to f(x) at x=0, wehave to chose the coefficients a0 through an properly. How do we do this?Well write down the polynomial and its derivatives as follows.     fn(x)= a0 + a1x + a2x2 + a3x3 + ... + anxnf1n(x)= a1 + 2a2x + 3a3x2 +... + nanxn-1f2n(x)= 2a2 + 6a3x +... +n(n-1)anxn-2     .     .f(n)n(x)= (n)an     Next we will substitute 0 in for x above so thata0=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) atx=0.     fn(x)= f(0) + f1(0)x + f2(0)x2/2 + ... + f(n)(0)xn/ n     This equation is called the nth head Taylor polynomial at x=0.Furthermore, we can generalize this equation for x=a instead of just