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The purpose of this lab is to use Maple to introduce you to Taylor polynomial approximations to functions, including some applications.

The general form of the Taylor polynomial approximation of order *n*
to *f*(*x*) is given by the following

**Theorem 1**

Suppose that *f*(*x*) is a smooth function in some open interval
containing *x*=*a*. Then the *n*th degree Taylor polynomial of the
function *f*(*x*) at the point *x*=*a* is given by

We will be seeing this formula a lot, so it
would be good for you to memorize it now! The notation
*f*^{(k)}(*a*) is used in the definition to stand for the value of the
*k*-th derivative of *f* at *x*=*a*. That is, *f ^{(1)}*(

**Theorem 2**

Suppose *f*(*x*) is a smooth function in some open interval
containing *x*=*a* and that *k* is a positive integer. Then *g*_{k}(*x*) defined by

*g*_{k}(*a*) = 0

*g*_{k}^{(k)}(*a*) = *f*^{(k)}(*a*)

- 1.
- For the following functions and base points, determine what
order is required so that the Taylor polynomial approximates the
function to within a tolerance of 0.5 over the given
interval. Include explanations how you got your answers in your report.
- (a)
- , base point
*x*=0, interval [-2,2]. - (b)
- , base point
*x*=0, interval . - (c)
- , base point
*x*=0, interval [-0.9,0.9]. - (d)
*f*(*x*) = 1/*x*, base point*x*=1, interval [0.5, 1.5].

- 2.
- Prove Theorem 2. Remember that an example is not a proof! (Note - this should be done by hand. part of using Maple is knowing when not to use it.)
- 3.
- For the third function in exercise 1, , consider the
Taylor polynomial with base point
*x*=0. Can you choose the order so that the Taylor polynomial is a good approximation (within 0.1, say) to at*x*=2? How about at*x*=0.5? Discuss the difference in the behavior of the Taylor polynomials at these two points. Can you divide the real line up into two parts, one where the approximation is good and one where it is bad? - 4.
- You know that the derivative of is . Can you
find a relationship between the derivative of the order
Taylor polynomial with base point
*x*=0 for and and a Taylor polynomial of some order with base point*x*=0 for ? - 5.
- You know that the indefinite integral of is
. Can you find a relationship between the indefinite
integral of the order
Taylor polynomial with base point
*x*=0 for and and a Taylor polynomial of some order with base point*x*=0 for ? (Hint - your answer should be yes.) - 6.
- One of the applications of Taylor polynomials is in approximating integrals. In the last lab, we encountered the error function , which is defined by The idea of using Taylor polynomials to approximate the integral is to replace the with its Taylor polynomial, which can be easily integrated. Can you find the minimum order required to approximate to within an accuracy of 0.05? Use zero for the base point of the Taylor polynomials.

11/9/1998