LaGrange Remainder Infinite Series

[carlodelmundo]

Homework Statement

Let f be a function whose seventh derivative is f⁷(x) = 10,000cos x. If x = 1 is in the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at x = 0 will approximate f(1) with an error of not more than

a.) 2.45 x 10_-5

b.) 1.98 x 10^-4

c.) 3.21 x 10^-2

d.) 0.248

e.) 1.984

Homework Equations

The Lagrange Remainder Formula, it states that the biggest error is only as large as the next sum in the series.

The formula is:

R_n \leq\frac{f^{7}(z)(x-c)^7}{(n+1)!}

The Attempt at a Solution

The maximum error is the next term in the sequence. Looking at the lagrange formula, we're looking for the maximum error, or the f⁷(z) term. Since we are given f⁷(x) = 10,000cos x and given that x = 1 is in the interval of convergence... I assumed that f⁷(1) = 10,000cos(1) is the maximum error.

Plugging it back in the LaGrange Formula I get the following:

R₆ \leq \frac{f^{7}(z)x^7}{7!} = \frac{10000cos 1}{7!}

I get a number that is not in the multiple choice answers. Any tips/ideas? Thanks

 

[HallsofIvy]

Homework Statement

Let f be a function whose seventh derivative is f⁷(x) = 10,000cos x. If x = 1 is in the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at x = 0 will approximate f(1) with an error of not more than

a.) 2.45 x 10_-5

b.) 1.98 x 10^-4

c.) 3.21 x 10^-2

d.) 0.248

e.) 1.984

Homework Equations

The Lagrange Remainder Formula, it states that the biggest error is only as large as the next sum in the series.

The formula is:

R_n \leq\frac{f^{7}(z)(x-c)^7}{(n+1)!}

The Attempt at a Solution

The maximum error is the next term in the sequence. Looking at the lagrange formula, we're looking for the maximum error, or the f⁷(z) term. Since we are given f⁷(x) = 10,000cos x and given that x = 1 is in the interval of convergence... I assumed that f⁷(1) = 10,000cos(1) is the maximum error.

Why would you assume that? Cosine is a decreasing function between 0 and 1< \pi/2. It takes its largest value at the lowest value of x in the interval, not the highest.

Plugging it back in the LaGrange Formula I get the following:

R₆ \leq \frac{f^{7}(z)x^7}{7!} = \frac{10000cos 1}{7!}

I get a number that is not in the multiple choice answers. Any tips/ideas? Thanks

 

[carlodelmundo]

I see my error. What threw me off is at the beginning of the problem how "1" is in the interval of convergence. I immediately thought that f⁷ (1) was the maximum value.

We're just looking at the maximum value of f(x) = 10,000cos(x) correct? and cosine is between -1 and 1... so the maximum value is cos(0).

The lagrange remainder would be 10,000 / 7!.

Is this correct?