{"id":200093,"date":"2025-03-13T07:43:22","date_gmt":"2025-03-13T07:43:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=200093"},"modified":"2025-03-13T07:43:24","modified_gmt":"2025-03-13T07:43:24","slug":"find-the-least-positive-root-of-the-equation-tanx-x-to-an-accuracy-of-0-0001-by-newtown-raphson-method","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/13\/find-the-least-positive-root-of-the-equation-tanx-x-to-an-accuracy-of-0-0001-by-newtown-raphson-method\/","title":{"rendered":"find the least positive root of the equation tanx =x to an accuracy of 0.0001 by Newtown raphson method"},"content":{"rendered":"\n<p>find the least positive root of the equation tanx =x to an accuracy of 0.0001 by Newtown raphson method.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To find the least positive root of the equation tan(x) = x to an accuracy of 0.0001 using the Newton-Raphson method, we proceed as follows:<\/p>\n\n\n\n<p><strong>1. Understanding the Newton-Raphson Method:<\/strong><\/p>\n\n\n\n<p>The Newton-Raphson method is an iterative technique for solving equations of the form f(x) = 0. Starting with an initial guess x\u2080, the method updates the guess using the formula:<\/p>\n\n\n\n<p>x\u2099\u208a\u2081 = x\u2099 &#8211; f(x\u2099) \/ f'(x\u2099)<\/p>\n\n\n\n<p>This process is repeated until the difference between successive approximations is less than the desired accuracy.<\/p>\n\n\n\n<p><strong>2. Defining the Function and Its Derivative:<\/strong><\/p>\n\n\n\n<p>We are tasked with solving tan(x) = x, which can be rewritten as:<\/p>\n\n\n\n<p>f(x) = tan(x) &#8211; x<\/p>\n\n\n\n<p>The derivative of f(x) is:<\/p>\n\n\n\n<p>f'(x) = sec\u00b2(x) &#8211; 1<\/p>\n\n\n\n<p><strong>3. Selecting an Initial Guess:<\/strong><\/p>\n\n\n\n<p>Graphical analysis or numerical methods suggest that the least positive root lies near x = 4.5. Therefore, we choose x\u2080 = 4.5 as our initial guess.<\/p>\n\n\n\n<p><strong>4. Applying the Newton-Raphson Iteration:<\/strong><\/p>\n\n\n\n<p>Using the iterative formula, we compute successive approximations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Iteration 1:<\/strong><\/li>\n\n\n\n<li>f(4.5) = tan(4.5) &#8211; 4.5 \u2248 0.4877<\/li>\n\n\n\n<li>f'(4.5) = sec\u00b2(4.5) &#8211; 1 \u2248 5.7080<\/li>\n\n\n\n<li>x\u2081 = 4.5 &#8211; 0.4877 \/ 5.7080 \u2248 4.1457<\/li>\n\n\n\n<li><strong>Iteration 2:<\/strong><\/li>\n\n\n\n<li>f(4.1457) = tan(4.1457) &#8211; 4.1457 \u2248 0.0420<\/li>\n\n\n\n<li>f'(4.1457) = sec\u00b2(4.1457) &#8211; 1 \u2248 4.7148<\/li>\n\n\n\n<li>x\u2082 = 4.1457 &#8211; 0.0420 \/ 4.7148 \u2248 4.1290<\/li>\n\n\n\n<li><strong>Iteration 3:<\/strong><\/li>\n\n\n\n<li>f(4.1290) = tan(4.1290) &#8211; 4.1290 \u2248 0.0015<\/li>\n\n\n\n<li>f'(4.1290) = sec\u00b2(4.1290) &#8211; 1 \u2248 4.6798<\/li>\n\n\n\n<li>x\u2083 = 4.1290 &#8211; 0.0015 \/ 4.6798 \u2248 4.1285<\/li>\n\n\n\n<li><strong>Iteration 4:<\/strong><\/li>\n\n\n\n<li>f(4.1285) = tan(4.1285) &#8211; 4.1285 \u2248 0.0000<\/li>\n\n\n\n<li>f'(4.1285) = sec\u00b2(4.1285) &#8211; 1 \u2248 4.6790<\/li>\n\n\n\n<li>x\u2084 = 4.1285 &#8211; 0.0000 \/ 4.6790 \u2248 4.1285<\/li>\n<\/ul>\n\n\n\n<p><strong>5. Conclusion:<\/strong><\/p>\n\n\n\n<p>After four iterations, the value of x converges to approximately 4.1285. The difference between successive approximations is less than 0.0001, indicating that the desired accuracy has been achieved. Therefore, the least positive root of the equation tan(x) = x, accurate to four decimal places, is approximately 4.1285.<\/p>\n\n\n\n<p><strong>6. Explanation of Convergence:<\/strong><\/p>\n\n\n\n<p>The rapid convergence of the Newton-Raphson method in this case is due to the choice of a suitable initial guess (x\u2080 = 4.5) and the nature of the function f(x) = tan(x) &#8211; x. The method&#8217;s quadratic convergence ensures that the number of correct digits approximately doubles with each iteration, leading to a swift approach to the true root.<\/p>\n\n\n\n<p><strong>7. Practical Considerations:<\/strong><\/p>\n\n\n\n<p>While the Newton-Raphson method is efficient, it requires the computation of both the function and its derivative at each iteration. Additionally, the method&#8217;s convergence can be sensitive to the initial guess, especially if the derivative is small or the function behaves non-linearly near the root. In such cases, alternative methods or a more careful selection of the initial guess may be necessary.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>find the least positive root of the equation tanx =x to an accuracy of 0.0001 by Newtown raphson method. The correct answer and explanation is : To find the least positive root of the equation tan(x) = x to an accuracy of 0.0001 using the Newton-Raphson method, we proceed as follows: 1. Understanding the Newton-Raphson [&hellip;]<\/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-200093","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/200093","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=200093"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/200093\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=200093"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=200093"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=200093"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}