{"id":191054,"date":"2025-02-14T05:46:55","date_gmt":"2025-02-14T05:46:55","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191054"},"modified":"2025-02-14T05:46:57","modified_gmt":"2025-02-14T05:46:57","slug":"draw-the-graph-of-%c6%92x-sinx2-2-in-the-viewing-rectangle-0-1-by-0-0-5-and-let-width","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/14\/draw-the-graph-of-%c6%92x-sinx2-2-in-the-viewing-rectangle-0-1-by-0-0-5-and-let-width\/","title":{"rendered":"Draw the graph of \u0192(x) = sin(x2\/2) in the viewing rectangle [0, 1] by [0, 0.5] and let width"},"content":{"rendered":"\n<p>Draw the graph of \u0192(x) = sin(x2\/2) in the viewing rectangle [0, 1] by [0, 0.5] and let width=.<\/p>\n\n\n\n<p>(a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate I.<\/p>\n\n\n\n<p>(b) For any value of , list the numbers Ln, Rn, Mn, Tn, and I in increasing order.<\/p>\n\n\n\n<p>(c) Compute L5, R5, M5, and T5. From the graph, which do you think gives the best estimate of I?<\/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<h3 class=\"wp-block-heading\">(a) Overestimation and Underestimation Analysis<\/h3>\n\n\n\n<p>To determine whether the left sum (L_2), right sum (R_2), midpoint sum (M_2), and trapezoidal sum (T_2) overestimate or underestimate the integral:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Left sum ( L_n )<\/strong>: Uses left endpoints; since ( f(x) ) is increasing on ([0,1]), ( L_n ) <strong>underestimates<\/strong> the integral.<\/li>\n\n\n\n<li><strong>Right sum ( R_n )<\/strong>: Uses right endpoints; since ( f(x) ) is increasing, ( R_n ) <strong>overestimates<\/strong> the integral.<\/li>\n\n\n\n<li><strong>Midpoint sum ( M_n )<\/strong>: Uses midpoints; provides a more balanced approximation.<\/li>\n\n\n\n<li><strong>Trapezoidal sum ( T_n )<\/strong>: Averages ( L_n ) and ( R_n ); typically more accurate.<\/li>\n<\/ul>\n\n\n\n<p>Thus:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( L_2 ) underestimates.<\/li>\n\n\n\n<li>( R_2 ) overestimates.<\/li>\n\n\n\n<li>( M_2 ) provides a more accurate estimate.<\/li>\n\n\n\n<li>( T_2 ) refines the estimate between ( L_2 ) and ( R_2 ).<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(b) Order of Values for Any ( n )<\/h3>\n\n\n\n<p>For any ( n ), the relationship follows:<\/p>\n\n\n\n<p>[<br>L_n \\leq M_n \\leq I \\leq T_n \\leq R_n<br>]<\/p>\n\n\n\n<p>where ( I ) represents the actual integral value.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(c) Computation of ( L_5, R_5, M_5, T_5 )<\/h3>\n\n\n\n<p>We will compute these sums numerically.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Results for ( n = 5 ):<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( L_5 = 0.1187 )<\/li>\n\n\n\n<li>( R_5 = 0.2146 )<\/li>\n\n\n\n<li>( M_5 = 0.1622 )<\/li>\n\n\n\n<li>( T_5 = 0.1666 )<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Best Estimate of ( I ):<\/h4>\n\n\n\n<p>From the function\u2019s behavior and past numerical analysis, the <strong>midpoint sum<\/strong> ( M_n ) typically provides the most accurate approximation for definite integrals because it accounts for concavity better than left or right sums. However, the <strong>trapezoidal rule<\/strong> ( T_n ), being an average of ( L_n ) and ( R_n ), often performs well too.<\/p>\n\n\n\n<p>Given the results, ( M_5 ) (0.1622) and ( T_5 ) (0.1666) are close, with ( T_5 ) likely being a slightly better approximation of ( I ). The <strong>trapezoidal rule<\/strong> generally balances out the overestimation from ( R_n ) and the underestimation from ( L_n ), making it a preferred choice.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (300 Words)<\/h3>\n\n\n\n<p>Numerical integration approximations like <strong>Riemann sums and the trapezoidal rule<\/strong> are essential for estimating definite integrals when an exact antiderivative is difficult to compute. Given ( f(x) = \\sin(x^2\/2) ), which is <strong>increasing<\/strong> on ([0,1]), we analyze different sum approximations.<\/p>\n\n\n\n<p>The <strong>left Riemann sum<\/strong> (( L_n )) underestimates the integral because it always takes function values at the beginning of each subinterval, missing the increase in function value. The <strong>right Riemann sum<\/strong> (( R_n )) overestimates because it takes function values at the right endpoints, capturing more of the higher function values.<\/p>\n\n\n\n<p>The <strong>midpoint rule<\/strong> (( M_n )) provides an improved estimate because it uses values at the center of each subinterval, better averaging over the curve&#8217;s behavior. The <strong>trapezoidal rule<\/strong> (( T_n )) is even more refined, averaging ( L_n ) and ( R_n ) to provide a balanced approximation.<\/p>\n\n\n\n<p>For ( n = 5 ), we find:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( L_5 = 0.1187 ) (underestimate)<\/li>\n\n\n\n<li>( R_5 = 0.2146 ) (overestimate)<\/li>\n\n\n\n<li>( M_5 = 0.1622 ) (closer to actual integral)<\/li>\n\n\n\n<li>( T_5 = 0.1666 ) (best overall estimate)<\/li>\n<\/ul>\n\n\n\n<p>Between ( M_5 ) and ( T_5 ), the <strong>trapezoidal rule provides the best estimate<\/strong> of the integral because it balances overestimation and underestimation. If a higher accuracy is needed, increasing ( n ) (subintervals) further refines the approximation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Draw the graph of \u0192(x) = sin(x2\/2) in the viewing rectangle [0, 1] by [0, 0.5] and let width=. (a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate I. (b) For any value of , list the numbers Ln, Rn, Mn, Tn, and I in increasing order. (c) Compute [&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-191054","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191054","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=191054"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191054\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191054"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191054"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}