Graphs of Taylor Polynomials from Today’s Class

Click on the functions below to see the graphs shown in class today.

f(x) = \sqrt{x} near x=4

f(x) = e^x near x=0

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2 Responses to Graphs of Taylor Polynomials from Today’s Class

  1. I happened to came across with this particular problem on WeBWorK in which we haven’t yet cover in class:

    “Find T_{5}(x), the degree 5 Taylor polynomial of the function f(x)= cos(x) at a = 0. Find all values of x for which this approximation is within 0.00179 of the right answer. Assume for simplicity that we limit ourselves to |x| =< 1. "

    What are some of the ways that I can possibly use to approach the second part of this question?

  2. Kate Poirier says:

    Hi Yong Chen. We’ve noticed a lot of times that for x-values close to a, any Taylor polynomial T_n(x) approximates f(x). For example, f(a) = T_n(a) for any degree n. But for x-values far from a, T_n(x) may be very far away from f(x). The higher the degree, though, the better the approximation…which means that for x-values further from a, T_n(x) gets closer to f(x) as n gets larger.

    Here you’re trying to say something very specific about how well T_5(x) actually does approximate f(x). For x=0, then T_5(x) is certainly within 0.00179 of f(x) because T_5(0) = f(0). For x-values close to 0, T_5(x) will be close to f(x). What you’re trying to find out is exactly how far x can be from 0 before the difference between T_5(x) and f(x) becomes greater than 0.00179.

    There are several different approaches, but I think the best ones are all graphical. Here’s one suggestion. Use a graphing tool to graph the function f(x)-T_5(x). (Since f(0)=T_5(0), this graph will go through the origin.) Next graph two constant functions, one at height -0.00179 and one at height 0.00179. Now the question becomes: for which values of x is the graph of f(x)-T_5(x) in between the graphs of these two constant functions?

    Notice that this is actually a precalculus type of question. Forget that one of your functions is a Taylor polynomial. You’re given two continuous functions. You know that they agree for at least one value of x. Then you’re asked how close x has to be to that value for the two functions to be within 0.00179 of each other. Your book does talk about the error of a Taylor polynomial in section 8.4, which you can go ahead and read; it will say roughly what I just said in different words but let me emphasize again, that it’s nothing too new or fancy.

    The last statement in the question, where you’re told to assume that |x|<=1, actually has very little to do with the question itself and is just WebWork being very careful. Everyone has a different question so you'll all have different answers; if your answer for |x| is that it has to be less than 1, enter that number. If you get a number greater than 1, just enter 1. (It's possible that for some large values of |x|, your T_5(x) and f(x) will be close to one another again, but it'd esentially be a coincidence so WebWork doesn't care about those values.)

    tl;dr answer: graph everything.

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