{"id":1333,"date":"2020-03-21T07:17:12","date_gmt":"2020-03-21T07:17:12","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1333"},"modified":"2020-03-24T07:50:44","modified_gmt":"2020-03-24T07:50:44","slug":"ibdp-past-year-exam-questions-introduction-to-differential-calculus","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/introduction-to-differential-calculus\/ibdp-past-year-exam-questions-introduction-to-differential-calculus\/","title":{"rendered":"IBDP Past Year Exam Questions \u2013 Introduction to Differential Calculus"},"content":{"rendered":"<p><strong>1. [M15\/P1\/TZ1]<\/strong><\/p>\r\n<p><strong>(a)<\/strong> Expand (x + h)<sup>3<\/sup> .\u00a0 \u00a0[2 marks]<br \/>\r\n<strong>(b)<\/strong> Hence find the derivative of f (x) = x<sup>3<\/sup> from first principles.\u00a0 \u00a0[3 marks]<\/p>\r\n<div id=\"link1-link-1333\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1333, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1333\">Solution<\/span><\/a><\/div><div id=\"link1-content-1333\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1338\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-10.jpg\" alt=\"\" width=\"1000\" height=\"345\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-10.jpg 1000w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-10-300x104.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/1-10-768x265.jpg 768w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>2. [M16\/P1\/TZ2]<\/strong><\/p>\r\n<p>The function\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><\/math>\r\n is defined as \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><mo>=<\/mo><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><\/math>\r\n where\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>,<\/mo><mo>&#160;<\/mo><mi>b<\/mi><mo>,<\/mo><mo>&#160;<\/mo><mi>c<\/mi><mo>&#8712;<\/mo><mi mathvariant=\"normal\">&#8477;<\/mi><\/math>\r\n .<br \/>\r\nHayley conjectures that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mi>f<\/mi><mfenced><msub><mi>x<\/mi><mn>2<\/mn><\/msub><\/mfenced><mo>&#8211;<\/mo><mi>f<\/mi><mfenced><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mfenced><\/mrow><mrow><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>&#8211;<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>f<\/mi><mo>&#8216;<\/mo><mfenced><msub><mi>x<\/mi><mn>2<\/mn><\/msub><\/mfenced><mo>+<\/mo><mi>f<\/mi><mo>&#8216;<\/mo><mfenced><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mfenced><\/mrow><mn>2<\/mn><\/mfrac><mo>,<\/mo><mo>&#160;<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>&#8800;<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/math>\r\n <br \/>\r\n<br \/>\r\nShow that Hayley\u2019s conjecture is correct.\u00a0 \u00a0[6 marks]<\/p>\r\n<div id=\"link2-link-1333\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1333, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1333\">Solution<\/span><\/a><\/div><div id=\"link2-content-1333\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1341\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-8.jpg\" alt=\"\" width=\"1001\" height=\"587\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-8.jpg 1001w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-8-300x176.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/2-8-768x450.jpg 768w\" sizes=\"auto, (max-width: 1001px) 100vw, 1001px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>3. [M12\/P1\/TZ2]<\/strong><\/p>\r\n<p>Using the definition of a derivative as\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mo>&#8216;<\/mo><mfenced><mi>x<\/mi><\/mfenced><mo>=<\/mo><munder><mrow><munder><mrow><mi>l<\/mi><mi>i<\/mi><mi>m<\/mi><\/mrow><mrow><mi>h<\/mi><mo>&#8594;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfenced><mfrac><mrow><mi>f<\/mi><mfenced><mrow><mi>x<\/mi><mo>+<\/mo><mi>h<\/mi><\/mrow><\/mfenced><mo>&#8211;<\/mo><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><\/mrow><mi>h<\/mi><\/mfrac><\/mfenced><\/mrow><mrow><\/mrow><\/munder><\/math>\r\n , show that the derivative of \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math>\r\n is \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>&#8211;<\/mo><mn>2<\/mn><\/mrow><msup><mfenced><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfenced><mn>2<\/mn><\/msup><\/mfrac><\/math>\r\n .\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[4 marks]<\/p>\r\n<div id=\"link3-link-1333\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1333, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1333\">Solution<\/span><\/a><\/div><div id=\"link3-content-1333\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1345\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-6.jpg\" alt=\"\" width=\"883\" height=\"447\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-6.jpg 883w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-6-300x152.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/3-6-768x389.jpg 768w\" sizes=\"auto, (max-width: 883px) 100vw, 883px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p><strong>4. [N18\/P2\/TZ0]<\/strong><\/p>\r\n<p>Differentiate from first principles the function\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><mo>=<\/mo><mn>3<\/mn><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>&#8211;<\/mo><mi>x<\/mi><\/math>\r\n .\u00a0 \u00a0[5 marks]<\/p>\r\n<div id=\"link4-link-1333\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1333, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1333\">Solution<\/span><\/a><\/div><div id=\"link4-content-1333\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1419\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-7.jpg\" alt=\"\" width=\"1027\" height=\"1155\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-7.jpg 1027w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-7-267x300.jpg 267w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-7-911x1024.jpg 911w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/03\/4-7-768x864.jpg 768w\" sizes=\"auto, (max-width: 1027px) 100vw, 1027px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>&nbsp;<\/p><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"1. [M15\/P1\/TZ1] (a) Expand (x + h)3 .\u00a0 \u00a0[2 marks] (b) Hence find the derivative of f (x) = x3 from first principles.\u00a0 \u00a0[3 marks] 2. [M16\/P1\/TZ2] The function\u00a0 f is defined as fx=ax2+bx+c where\u00a0 a,&#160;b,&#160;c&#8712;&#8477; . Hayley conjectures that\u00a0 fx2&#8211;fx1x2&#8211;x1=f&#8216;x2+f&#8216;x12,&#160;x2&#8800;x1 Show that Hayley\u2019s conjecture is correct.\u00a0 \u00a0[6 marks] 3. [M12\/P1\/TZ2] Using the definition of [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1333","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-introduction-to-differential-calculus","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1333","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1333"}],"version-history":[{"count":15,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1333\/revisions"}],"predecessor-version":[{"id":1421,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1333\/revisions\/1421"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}