Answer the following questions. Everything must be done using the Maple. No computations on paper.

. Find the Taylor polynomials of degree 3, 5, and 8 for the following functions about the given points. Then plot a sequence of Taylor polynomials from 3 to 8 with the function on the same set of axes. Adjust your output to show that the Taylor polynomials get closer to the function as the degree increases, but sometimes only over a small piece of the x-axis.

1) cos(x) about x = 0. (Try x from –Pi to Pi)

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2) exp(x) about x = 0. (Use another small interval about 0)

3) square root (1+x) about x = 0. (Use an interval from -1 to 5. Notice the approximations are only close from -1 to 1.)

4) arctan(x) about x = 0. (Again, the approximations are close just from -1 to 1.)

5) ln(x) about x = 1. (Remember to change the Taylor Polynomial formula. Where are the approximations close to the function this time?)

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