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

Gods Gift to Calculators The Taylor Series     It is incredible how far calculators present come since my p atomic number 18nts were incollege, which was when the square root key came bring out. Calculators since thenhave 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 intravenous feeding basic operations addition, subtraction, multiplication, and division Sowhat is it that allows calculators to evaluate logs, trigonometric functions,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.  &n bsp  So how do mathematicians take a function and strain it into a polynomialfunction? Lets find out. First, lets assume that we have 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 using 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 will also pass through x=0, and will have the same slope as f(x) if we let a0=f1(0).      in a flash, if we want to propose 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) 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 degree Taylor polynomial at x=0.Furthermore, we can generalize this equation for x=a instead of just