Here are Taylor polynomials for f(x) = 1 centered at x = c. You can move the center point and you can change the Taylor

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Here Are Taylor Polynomials For F X 1 Centered At X C You Can Move The Center Point And You Can Change The Taylor 1 (72.75 KiB) Viewed 55 times

Here are Taylor polynomials for f(x) = 1 centered at x = c. You can move the center point and you can change the Taylor polynomial degree. Click here to view the graph in a separate window. JAKGraph v0.94 Copyright (C) see http://jsxgraph.org n=0 a. Set the center point at x = 2. Previous exploration discovered that there is an x-coordinate beyond which the Taylor polynomials fail to approximate f(x). (If you don't remember, explore again.) What is the distance from that location to the center point? b. Other examples, notably sin(x) and ex, had Taylor polynomials that did not display this behavior. For comparison, click here to look at sin(x). Move that slider around to see if there are any problem locations like the one you found above. Discussion Question: What suspicious feature does the function have that sin(x) does not have? X [Hint: It's "vertical"] What is the distance from this feature to the center point at x = 2? c. Move the center point to x = 5. Run the degree up to 80. What is the distance from the center to the point where approximations seem to fail? d. Keep the center at x = 5 and the degree at 80. What is the distance from the center to the feature that is likely causing the approximations to fail.

Here are Taylor polynomials for f(x) = In(x), centered at x = 1. Click here to view the graph in a separate window. JSXGraph v0.94 Copyright (C) see http://jsxgraph.org n=0 2.5 a. Is it possible to find a value of n such that T(x) is a good approximation at x = 1.5? O Yes O No b. Is it possible to find a value of n such that Tn(x) is a good approximation at x = 3? O Yes O No c. Is it possible to find a value of n such that Tn(x) is a good approximation at x = 2.2? O Yes O No d. Is it possible to find a value of n such that Tn(x) is a good approximation at x = 1.9? O Yes O No Discussion Question: It appears that approximations, for any size n, go from good to bad at about x = 2. This did not happen for ex or sin(x). What is different about In(x)? And why does x = 2 matter?