Taylor Polynomials

Prerequisites: Higher order derivatives

Goals: To become familiar with the concept of polynomials converging to a function. To conjecture the MacLaurin series for ex and cos(x) and the Taylor series for ln(x) centered at 1.

  1. Let's imagine that ex can be written as a polynomial (actually, it cannot, but that doesn't stop us from imagining that it can). Then

    ex=c0 +c1x+c2x2+c3x3+...+cnxn

    for some appropriate integer n≥0. For the moment, let's pretend that n=0, so ex=c0. Focusing on what is happening to ex at x=0, we get c0=___. Try graphing both (that is, ex and the value you got for c0) in Maple by using the code below. You will need to replace the blank with the value you got for c0.
    restart:with(plots):
    f:=x->___:
    expPlot1:=plot(exp(x),x=-3..3,y=-1..7,color=red):
    expPlot2:=plot(f(x),x=-3..3,y=-1..7,color=blue):
    display(expPlot1,expPlot2);
  2. They are clearly not the same functions, so let's move next to n=1. Then since ex=c0+c1x, we expect that their derivatives are also equal. Differentiate both sides and plug in x=0 to find c1=___. Again try graphing both (that is, ex and c0+c1x) by substituting your value of c0+c1x into the blank in the code (above) that you pasted into Maple. What do you notice?
  3. Again, they are clearly not the same functions, so let's try n=2. Then ex=c0 +c1x+c2x2. We already know that their zeroth and first derivatives are equal, so set their second derivatives equal and find c2=___. Again graph both with Maple. What do you notice?
  4. Repeat the procedure for n=3 to find c3=___, and describe the relationship of the two graphs.
  5. Repeat the procedure for n=4 to find c4=___, and describe the relationship of the two graphs.
  6. Find a general formula for cn=___. What do you expect to happen to the graphs if we take larger and larger values of n?

  7. Repeat the procedure to find a polynomial of degree 0, 1, 2, 3, and 4 to estimate cos(x), graphing your polynomials with the function cos(x) at each step. The above Maple code will need some adjustment to work properly. For example, you will need to change the function in the first plot from exp(x) to cos(x), and should probably adjust the domain and range of each plot, too.

  8. Explain why this procedure won't work (at least not in exactly the same way) for ln(x).

  9. Let's imagine that ln(x) can written as a polynomial in x-1, that is

    ln(x)=c0 +c1(x-1)+c2(x-1)2+c3(x-1)3+...+cn(x-1)n

    for some appropriate integer n≥0. Repeat the previous procedure for n=0, 1, 2, 3, and 4 to estimate ln(x), but focus on what is happening at x=1 instead of x=0. Here the following code may be helpful.
    restart:with(plots):
    h:=x->___:
    logPlot1:=plot(ln(x),x=-1..3,y=-3..2,color=red):
    logPlot2:=plot(h(x),x=-1..3,y=-3..2,color=blue):
    display(logPlot1,logPlot2);
  10. Find a general formula for cn=___ in problem 9. What do you think happens to the graphs if we take larger and larger values of n?
Last Update: 2006/07/26