{"id":20207,"date":"2024-04-15T05:49:49","date_gmt":"2024-04-15T05:49:49","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20207"},"modified":"2024-04-15T05:49:49","modified_gmt":"2024-04-15T05:49:49","slug":"the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/","title":{"rendered":"The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}&#8217;\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) &#8211; {\\text{f}}\\left( {{\\text{x}} &#8211; {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} &#8211; {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. > 1% and < 5% B. < 0.1% C. > 0.1% and < 1% D. > 5%"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;> 1% and < 5%\" option2=\"< 0.1%\" option3=\"> 0.1% and < 1%\" option4=\"> 5%&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{(C)}}$.<\/p>\n<p>The exact value of the derivative of $f(x) = \\cos x$ at $x = \\frac{\\pi}{6}$ radian is $-\\frac{\\sqrt{3}}{2}$. The approximate value of the derivative using the given formula is $\\frac{\\cos \\left( \\frac{\\pi}{6} + 0.1 \\right) &#8211; \\cos \\left( \\frac{\\pi}{6} &#8211; 0.1 \\right)}{2 \\cdot 0.1} = -0.49999999999999994$. The percentage error is therefore $\\left( \\frac{-\\frac{\\sqrt{3}}{2} &#8211; -0.49999999999999994}{-\\frac{\\sqrt{3}}{2}} \\times 100 \\right) \\% = 0.06666666666666666$, which is greater than $0.1\\%$ but less than $1\\%$.<\/p>\n<p>Here is a step-by-step solution:<\/p>\n<ol>\n<li>The exact value of the derivative of $f(x) = \\cos x$ at $x = \\frac{\\pi}{6}$ radian is $-\\frac{\\sqrt{3}}{2}$.<\/li>\n<li>The approximate value of the derivative using the given formula is $\\frac{\\cos \\left( \\frac{\\pi}{6} + 0.1 \\right) &#8211; \\cos \\left( \\frac{\\pi}{6} &#8211; 0.1 \\right)}{2 \\cdot 0.1} = -0.49999999999999994$.<\/li>\n<li>The percentage error is therefore $\\left( \\frac{-\\frac{\\sqrt{3}}{2} &#8211; -0.49999999999999994}{-\\frac{\\sqrt{3}}{2}} \\times 100 \\right) \\% = 0.06666666666666666$, which is greater than $0.1\\%$ but less than $1\\%$.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;> 1% and < 5%\" option2=\"< 0.1%\" option3=\"> 0.1% and < 1%\" option4=\"> 5%&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20207","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}&#039;\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) - {\\text{f}}\\left( {{\\text{x}} - {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} - {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. &gt; 1% and &lt; 5% B. &lt; 0.1% C. &gt; 0.1% and &lt; 1% D. &gt; 5%<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}&#039;\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) - {\\text{f}}\\left( {{\\text{x}} - {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} - {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. &gt; 1% and &lt; 5% B. &lt; 0.1% C. &gt; 0.1% and &lt; 1% D. &gt; 5%\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;&gt; 1% and &lt; 5%&quot; option2=&quot;&lt; 0.1%&quot; option3=&quot;&gt; 0.1% and &lt; 1%&quot; option4=&quot;&gt; 5%&#8221; correct=&#8221;option1&#8243;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:49:49+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}'\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) - {\\text{f}}\\left( {{\\text{x}} - {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} - {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. > 1% and < 5% B. < 0.1% C. > 0.1% and < 1% D. > 5%","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/","og_locale":"en_US","og_type":"article","og_title":"The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}'\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) - {\\text{f}}\\left( {{\\text{x}} - {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} - {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. > 1% and < 5% B. < 0.1% C. > 0.1% and < 1% D. > 5%","og_description":"[amp_mcq option1=&#8221;> 1% and < 5%\" option2=\"< 0.1%\" option3=\"> 0.1% and < 1%\" option4=\"> 5%&#8221; correct=&#8221;option1&#8243;]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:49:49+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/","name":"The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}'\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) - {\\text{f}}\\left( {{\\text{x}} - {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} - {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. > 1% and < 5% B. < 0.1% C. > 0.1% and < 1% D. > 5%","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:49:49+00:00","dateModified":"2024-04-15T05:49:49+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-derivative-of-fx-cos-x-can-be-estimated-using-the-approximation-textfleft-textx-right-fractextfleft-textx-texth-right-textfle\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Engineering maths","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/"},{"@type":"ListItem","position":4,"name":"Calculus","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/calculus\/"},{"@type":"ListItem","position":5,"name":"The derivative of f(x) = cos x can be estimated using the approximation \\[{\\text{f}}&#8217;\\left( {\\text{x}} \\right) = \\frac{{{\\text{f}}\\left( {{\\text{x}} + {\\text{h}}} \\right) &#8211; {\\text{f}}\\left( {{\\text{x}} &#8211; {\\text{h}}} \\right)}}{{2{\\text{h}}}}.\\] The percentage error is calculated as \\[\\left( {\\frac{{{\\text{Exact value}} &#8211; {\\text{Approx value}}}}{{{\\text{Exact value}}}} \\times 100} \\right)\\] The percentage error in the derivative of f(x) at \\[{\\text{x}} = \\frac{\\pi }{6}\\] radian choosing h = 0.1 radian is A. > 1% and < 5% B. < 0.1% C. > 0.1% and < 1% D. > 5%"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=20207"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20207\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}