# Thread:

1. ## Upper Bound For Maclaurin Approximation

There is a homework problem that I don't even understand what they're asking. Can any of you help clarify what they mean?

f(x) = sin(x)

Find an upper bound for |[f(x)-[p9(x)]]|
-7 <= x <= 7

p9(x) is the nth order Maclaurin approximaxion to f

2. Originally Posted by soma
There is a homework problem that I don't even understand what they're asking. Can any of you help clarify what they mean?
Where should we start? Obviously this problem expects you to know what a "MacLaurin series" is. Do you know that? It is simply the Taylor's series at x= 0. Do you know what that is? If not look up "MacLaurin series" or "Taylor's series" in the index of your book.

And, I strongly suspect that in the section where this problem is given, there is a formula for the error you get in using cutting a McLaurin or Taylor's polynomial of degree n (that is, cutting off the McLaurin or Taylor's series at the nth power). It should look something like this:
$E\le \frac{M_{n+1}}{(n+1)!}x^{n+1}$ where " $M_{n+1}$" is an upper bound on the absolute value of the n+1 derivative of the function.

Since this problem asked about the "9th order MacLaurin series for sin(x)" for x between -7 and 7. Okay you need $M_{10}$. What is the 10th derivative of sin(x)? What is its largest value between -7 and 7 (radians: $2\pi$ is about 6.28 so this is from less than $2\pi$ to larger than $2\pi$).

3. Ok, so the tenth derivative of sin(x) is -sin(x), and the maximum bound of -sin(x) is when x = -pi/2

-sin(-pi/2) = 1

so

E <= [1/(9+1)!]*[(-pi/2)^(9+1)]
E <= .0000252020

Am I understanding this problem correctly?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.951570451259613, "perplexity_flag": "middle"}