The purpose of this lab is to use Maple to introduce you to quadratic polynomial approximations to functions. These will turn out to be specific examples of a general and very useful class of approximations known as Taylor polynomials.
Suppose is a smooth function, that is, it has derivatives of all orders. Examples of such functions are polynomials and the transcendental functions , , and the trigonometric functions and . Polynomials are simple to do calculations with, but it isn't so obvious how your calculator or a computer does calculations with functions like and . In this lab we will start to investigate a method for approximating smooth functions with polynomials. Calculators and computers actually use related, but more sophisticated, methods to approximate the values of transcendental and trigonometric functions.
To help you understand where the polynomial approximations come from, recall the linear approximation to at x=a, which had the following properties:
Recall that the linear approximation was the straight line tangent to . That is, it went through the point and had slope , the derivative of at x=a.
For example, suppose and we choose a=0. To compute the linear approximation to at x=0, we need two numbers: and . Plugging these two numbers into the formula gives the answer
If you plotted the linear approximation on the same graph as the function , you would see that the linear approximation is tangent to the function at x=0.
Changing the value of the base point a results in a different linear function. For example, setting the value produces the linear approximation
which is tangent to at .
The linear approximation is usually a pretty good approximation to the function for values of x that are very close to the base point a. It often happens, though, that a better approximation is needed. One approach that usually works is to use a higher order polynomial, for example a quadratic polynomial, to approximate the function .
A way of defining a quadratic polynomial that builds on our definition of the linear approximation is given below. Simply put, we require that the quadratic polynomial go through the point , and have the same first and second derivatives as at the point x=a. Polynomial approximations defined in this way turn out to be very useful in a lot of different applications, and are referred to as Taylor polynomials.
Given a function and a base point a, the quadratic Taylor polynomial approximation can always be obtained by substituting into the three equations in the definition above and solving for the values of the coefficients A, B, and C. For example, suppose and we choose . Then subsituting into the three equations in the definition produces the following equations.
Solving these equations is straightforward; just start with the third equation and work your way back, leading to the solution
Maple commands for solving this example and plotting the result are shown below. Note the use of D@D to generate the second derivatives. Maple uses the @ symbol to represent composition.
> p2 := x->A*x^2+B*x+C;
> f := x -> cos(x);
> a := Pi;
> sol1 := solve(f(a)=p2(a),D(f)(a)=D(p2)(a),(D@D)(f)(a)=(D@D)(p2)(a),A,B,C);
> f2 := x -> subs(sol1,p2(x));
The definition above can always be used to find the quadratic Taylor polynomial approximation to a smooth function, but it turns out that if we write the polynomial in powers of , there is a useful general formula, which is given in the following theorem.
For example, using the formula in the theorem, we would write the quadratic Taylor polynomial approximation to with base point as
The general formula may seem a little cumbersome at first, but you need to become familiar with it. Note that x is the independent variable in the formula, while a is a parameter that is either a constant or can be treated as one. This means that things like and should be treated as constants.
Showing that the formula in the theorem satisfies the definition is not difficult if you remember that x is the variable and a is a parameter. Keeping this in mind, the first and second derivatives of the formula in the theorem are and . Given the derivatives, showing that the formula in the theorem satisfies the definition is simply a matter of substituting x=a.
The CalcP package has a command called Taylor that can be used to generate quadratic Taylor polynomial approximations, as well as the higher-order Taylor polynomial approximations that we'll study later. (If you look at the help page for Taylor, you'll find out that the third argument determines the order of the Taylor polynomial generated. This argument is 2 in the examples because we are generating the quadratic polynomials.) The Maple commands given below provide examples of how to use this command in generating, plotting, and manipulating quadratic Taylor polynomials. The last example shows how to plot the absolute value of the error, defined as , of a Taylor polynomial approximation.
> plot(f(x),Taylor(f(x),x=Pi,2), x=0..2*Pi);
> fquad := x-> Taylor(f(x),x=Pi,2) ;