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

Gods Gift to Calculators The Taylor Series     It is incredible how far calculators contain come since my pargonnts were incollege, which was when the square root key came out. Calculators since thenhave evolved into machines that can realise natural logarithms, sines, cosines,arcsines, and so on. The funny thing is that calculators have not gotten every"smarter" since then. In fact, calculators are still prefatorialally limited to the quartet basic operations addition, subtraction, multiplication, and division Sowhat is it that allows calculators to evaluate logs, trigonometric becomes,and exponents? This ability is cod in large part to the Taylor series, whichhas allowed mathematicians (and calculators) to approximate functions,such asthose given above, with multinomials. These polynomials, called TaylorPolynomials, are easy for a calculator manipulate because the calculator usesonly the four basic arithmetic operators.     So h ow do mathematicians take a function and turn it into a polynomialfunction? Lets find out. First, allows assume that we have a function in the formy= f(x) that looks like the graphical record below.     Well start out trying to approximate function values stuffy x=0. To dothis we start out using the lowest put in 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 will in like manner pass through x=0, and will have the same hawk as f(x) if we let a0=f1(0).      at a time, if we want 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 match the the derivatives of f(x) atx=0. So if we want to make the derivatives of fn(x) rival to f(x) at x=0, wehave to chose the coefficients a0 through an properly. How do we do this?Well publish 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 degree Taylor polynomial at x=0.Furthermore, we can generalize this equation for x=a instead of just