Lab 13

MAT 229, Spring 2021

Review

The (n^th) degree Taylor polynomial of f(x) centered at a is the partial sum of the Taylor series that goes up to and includes the (n^th) power of (x-a). If the Taylor series for f(x) centered at a converges to f(x) for a given value x, then the (n^th) Taylor polynomial of f(x) centered at a provides a polynomial approximation to f(x).

The (n^th) Taylor remainder of g(x) centered at a is

(Subscript[R, n])(x)=g(x)-(Underoverscript[Σ, (k=0), n])(((g^((k)))(a))/(k!))(((x-a))^k)

In other words the error in using the (n^th) degree Taylor polynomial to approximate the function is error=|(Subscript[R, n])(x)|

Analyzing error

The Taylor series error estimate: If |(f^((n+1)))(x)|≤M for all values of x of interest, then

|(Subscript[R, n])(x)|≤(M/((n+1)!))|x-a((|)^(n+1)).

You can think of this roughly as the error on the interval is smaller than the largest “first neglected term” on the interval.

Questions to submit

Instructions: Do your work on paper and submit as a pdf file. Show your work.

Let g(x)=sin(x).

What is the Taylor series centered at zero for g(x)?

What is a simple estimate for M in the remainder for this function? (|(g^((n+1)))(x)|≤M)

What degree Taylor polynomial for g(x) will approximate it with error less than 0.01 for |x|<1?

What degree Taylor polynomial for g(x) will approximate it with error less than 0.01 for |x|<2?

What degree Taylor polynomial for g(x) will approximate it with error less than 0.01 for |x|<3?

Let h(x)=ln(x).

What is the Taylor series centered at 1 for h(x)?

What is the interval of convergence for that Taylor series?

What is true about any estimate for M in the remainder for this function for the values of x in the interval of convergence?

What if we use the (5^th) degree Taylor polynomial, (Subscript[T, 5])(x), centered at 1 to approximate h(x) for |x-1|≤0.5? What is a good estimate for M?

Approximate ln(1.5) with (Subscript[T, 5])(1.5). What is an estimate of the error in this approximation?

Let f(x)=(e^x).

What is the Taylor series centered at zero for (e^x)?

If we plan to use an (n^th) degree Taylor polynomial, (Subscript[T, n])(x), to approximate f(x) for -1≤x≤1, what is an estimate for M?

Find a value of n so that (Subscript[T, n])(x) approximates f(x) with error less than 0.0001 for all x with |x|≤1.