{"id":185203,"date":"2025-01-22T08:11:46","date_gmt":"2025-01-22T08:11:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185203"},"modified":"2025-01-22T08:11:48","modified_gmt":"2025-01-22T08:11:48","slug":"find-the-taylor-series-expansion-for-fx-sin-2x-at-x-pi-write-the-series-with-at-least-three-nonzero-terms","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/22\/find-the-taylor-series-expansion-for-fx-sin-2x-at-x-pi-write-the-series-with-at-least-three-nonzero-terms\/","title":{"rendered":"Find the Taylor series expansion for f(x) =sin 2x at x = pi (write the series with at least three nonzero terms)"},"content":{"rendered":"\n

Find the Taylor series expansion for f(x) =sin 2x at x = pi (write the series with at least three nonzero terms). From part (a), find the 5^th degree polynomial P_5 (x) to approximate f(x). Approximate f(3) using P_5(x) above.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Step 1: Taylor Series Expansion<\/h3>\n\n\n\n

The Taylor series expansion of ( f(x) ) about ( x = a ) is:<\/p>\n\n\n\n

[
f(x) = \\sum_{n=0}^\\infty \\frac{f^{(n)}(a)}{n!}(x-a)^n
]<\/p>\n\n\n\n

For ( f(x) = \\sin(2x) ), expand at ( x = \\pi ).<\/p>\n\n\n\n

Derivatives:<\/h4>\n\n\n\n
    \n
  1. ( f(x) = \\sin(2x) ), so ( f(\\pi) = \\sin(2\\pi) = 0 ).<\/li>\n\n\n\n
  2. First derivative: ( f'(x) = 2\\cos(2x) ), so ( f'(\\pi) = 2\\cos(2\\pi) = 2 ).<\/li>\n\n\n\n
  3. Second derivative: ( f”(x) = -4\\sin(2x) ), so ( f”(\\pi) = -4\\sin(2\\pi) = 0 ).<\/li>\n\n\n\n
  4. Third derivative: ( f”'(x) = -8\\cos(2x) ), so ( f”'(\\pi) = -8\\cos(2\\pi) = -8 ).<\/li>\n\n\n\n
  5. Fourth derivative: ( f^{(4)}(x) = 16\\sin(2x) ), so ( f^{(4)}(\\pi) = 16\\sin(2\\pi) = 0 ).<\/li>\n\n\n\n
  6. Fifth derivative: ( f^{(5)}(x) = 32\\cos(2x) ), so ( f^{(5)}(\\pi) = 32\\cos(2\\pi) = 32 ).<\/li>\n<\/ol>\n\n\n\n

    Taylor Series:<\/h4>\n\n\n\n

    Using the formula:
    [
    f(x) = f(\\pi) + f'(\\pi)(x-\\pi) + \\frac{f”(\\pi)}{2!}(x-\\pi)^2 + \\frac{f”'(\\pi)}{3!}(x-\\pi)^3 + \\ldots
    ]<\/p>\n\n\n\n

    Substituting derivatives at ( x = \\pi ):
    [
    f(x) \\approx 0 + 2(x-\\pi) + 0 + \\frac{-8}{6}(x-\\pi)^3 + 0 + \\frac{32}{120}(x-\\pi)^5
    ]
    [
    f(x) \\approx 2(x-\\pi) – \\frac{4}{3}(x-\\pi)^3 + \\frac{4}{15}(x-\\pi)^5
    ]<\/p>\n\n\n\n

    Step 2: Fifth-Degree Polynomial ( P_5(x) )<\/h3>\n\n\n\n

    The fifth-degree polynomial is:
    [
    P_5(x) = 2(x-\\pi) – \\frac{4}{3}(x-\\pi)^3 + \\frac{4}{15}(x-\\pi)^5
    ]<\/p>\n\n\n\n

    Step 3: Approximation of ( f(3) )<\/h3>\n\n\n\n

    Substitute ( x = 3 ):
    [
    P_5(3) = 2(3-\\pi) – \\frac{4}{3}(3-\\pi)^3 + \\frac{4}{15}(3-\\pi)^5
    ]<\/p>\n\n\n\n

    Numerically calculating:
    [
    3 – \\pi \\approx -0.1416
    ]
    [
    P_5(3) \\approx 2(-0.1416) – \\frac{4}{3}(-0.1416)^3 + \\frac{4}{15}(-0.1416)^5
    ]
    [
    P_5(3) \\approx -0.2832 – 0.000377 + 0.000012
    ]
    [
    P_5(3) \\approx -0.2836
    ]<\/p>\n\n\n\n

    Explanation (300 Words)<\/h3>\n\n\n\n

    The Taylor series expands a function ( f(x) ) about a point ( x = a ), allowing approximation through polynomials. Each term depends on derivatives of ( f(x) ) evaluated at ( x = a ), ensuring local accuracy. For ( f(x) = \\sin(2x) ), we derive its Taylor series about ( x = \\pi ) using successive derivatives. Odd derivatives contribute non-zero terms, reflecting ( \\sin(2x) )’s periodic nature, while even derivatives vanish because ( \\sin(2\\pi) = 0 ).<\/p>\n\n\n\n

    The resulting series up to the fifth degree, ( P_5(x) ), includes terms up to ( (x-\\pi)^5 ). The series approximates ( f(x) ) closely near ( x = \\pi ), with error decreasing as higher-order terms are included.<\/p>\n\n\n\n

    To approximate ( f(3) ), substitute ( x = 3 ) into ( P_5(x) ). Numerical substitution involves calculating powers of ( (3-\\pi) \\approx -0.1416 ). The result, ( P_5(3) \\approx -0.2836 ), provides a close approximation to ( f(3) ), as higher-degree terms contribute minimally.<\/p>\n\n\n\n

    This method demonstrates how Taylor series simplify complex trigonometric functions to polynomials, useful in physics, engineering, and numerical computation where exact values are challenging to compute.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Find the Taylor series expansion for f(x) =sin 2x at x = pi (write the series with at least three nonzero terms). From part (a), find the 5^th degree polynomial P_5 (x) to approximate f(x). Approximate f(3) using P_5(x) above. The Correct Answer and Explanation is : Step 1: Taylor Series Expansion The Taylor series […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-185203","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=185203"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185203\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185203"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185203"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}