Inrtoducing Power Series 2

In our last post we found that we could produce better and better polynomial approximations to a function. That is, we produced a set of polynomials of increasing degree that had the same value for the functions and its derivatives at a given point. To see what is going on I suggest we graph these approximating polynomials along with the given function.

We found that the polynomials \left( x-1 \right), \left( x-1 \right)+\left( -\tfrac{1}{2} \right){{\left( x-1 \right)}^{2}}, \left( x-1 \right)+\left( -\tfrac{1}{2} \right){{\left( x-1 \right)}^{2}}+\left( \tfrac{1}{6} \right){{\left( x-1 \right)}^{3}}, and \left( x-1 \right)+\left( -\tfrac{1}{2} \right){{\left( x-1 \right)}^{2}}+\left( \tfrac{1}{6} \right){{\left( x-1 \right)}^{3}}+\left( -\tfrac{1}{4!} \right){{\left( x-1 \right)}^{4}} produced approximations to the natural logarithm function at the point (1, 0). To see how this works, graph each of these polynomials, one after the other. See the figure below.

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Notice that each polynomial comes closer to the graph of the graph of y = ln(x), the black graph, in the figures.

You students can do this on their graphing calculators or with a graphing program. More on how to do  this below.

Now do the same thing with the polynomials found for the sine function.

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However, there is a difference. The sine polynomials seem to hug the sine graph over increasingly wider intervals while the logarithm polynomials do not. This may not be a surprise since the logarithm function has no values for x\le 0 while the polynomials do. The polynomials cannot come close to the graph if there is no graph.

Students should notice these things:

  • Successively higher degree polynomials seem to come closer to the graph of the function than the previous one.
  • The polynomials may exist outside the domain of the function (outside of x>0 for ln(x) for example).
  • The interval where the graphs are near the function is limited.

Taken together these two examples suggest several questions (which you can perhaps draw out of your class):

  1. If there were an infinite number of terms would the Polynomial be the same as the function?
  2. How do you add an infinite number of terms?
  3. Over what interval is the approximation “good”? Is the interval the same for all functions? How do you find the interval?
  4. How good is the approximation?
  5. Is there an easier way to build the polynomial? Do you have to figure out and evaluate all of the derivatives?

These questions will be the topic of my next post.

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How to Graph these Polynomials using Winplot

You can enter each polynomial separately of course, but here is an easier way.

  1. After setting your viewing window (CTRL+V), push [F1] to get the explicit equation entry window and enter sin(x) (or the function you are interested in) and click [OK] to graph y = sin(x).
  2. Then push [F1] again and enter Sum( (-1)^(n+1)x^(2n-1)/(2n-1)! ,n,1,A) and click [OK]. The underlined part may be changed to the general term of any series.  The n identifies the variable, the 1 is the starting value of n and the A will be the final value which we will change.
  3. Next click on [ANIM] > [Individual] > [A]. This will bring up a slider. Enter 100 in the box and click [Set R] and then enter 0 and click [Set L]. This will make the A values change by exactly 1 allowing you to look at A = 1, 2, 3, 4, … in order.
  4. Click the tab on the “A” slider window box and see the various approximating polynomials “hug” the graph

 

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