{"id":1382,"date":"2020-03-22T15:34:35","date_gmt":"2020-03-22T15:34:35","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&p=1382"},"modified":"2020-03-26T10:46:27","modified_gmt":"2020-03-26T10:46:27","slug":"ibdp-past-year-exam-papers-maclauren-series","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/maclaurin-series\/ibdp-past-year-exam-papers-maclauren-series\/","title":{"rendered":"IBDP Past Year Exam Papers – Maclauren Series"},"content":{"rendered":"

1. [M17\/P3\/TZ0]<\/strong><\/p>\r\n

Let the Maclaurin series for \r\ntan<\/mi> <\/mo>x<\/mi><\/math>\r\n be\u00a0 \r\ntan<\/mi> <\/mo>x<\/mi>=<\/mo> <\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi> <\/mo>+<\/mo> <\/mo>a<\/mi>3<\/mn><\/msub>x<\/mi>3<\/mn><\/msup> <\/mo>+<\/mo> <\/mo>a<\/mi>5<\/mn><\/msub>x<\/mi>5<\/mn><\/msup> <\/mo>+<\/mo> <\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo><\/math>\r\n where\u00a0 \r\na<\/mi>1<\/mn><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>3<\/mn><\/msub> <\/mo><\/math>\r\n and\u00a0 \r\na<\/mi>5<\/mn><\/msub><\/math>\r\n are constants.<\/p>\r\n

(a)<\/strong> Find series for\u00a0 \r\ns<\/mi>e<\/mi>c<\/mi>2<\/mn><\/msup> <\/mo>x<\/mi><\/math>\r\n , in terms of\u00a0 \r\na<\/mi>1<\/mn><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>3<\/mn><\/msub><\/math>\r\n and\u00a0 \r\na<\/mi>5<\/mn><\/msub><\/math>\r\n , up to and including the\u00a0 \r\nx<\/mi>4<\/mn><\/msup><\/math>\r\n term<\/p>\r\n

(i) by differentiating the above series for\u00a0 \r\ntan<\/mi> <\/mo>x<\/mi><\/math>\r\n ;
\r\n(ii) by using the relationship\u00a0 \r\ns<\/mi>e<\/mi>c<\/mi>2<\/mn><\/msup> <\/mo>x<\/mi>=<\/mo> <\/mo>2<\/mn> <\/mo>+<\/mo> <\/mo>tan<\/mi>2<\/mn><\/msup> <\/mo>x<\/mi><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n

(b)<\/strong> Hence, by comparing your two series, determine the values of\u00a0 \r\na<\/mi>1<\/mn><\/msub> <\/mo>,<\/mo> <\/mo>a<\/mi>3<\/mn><\/msub><\/math>\r\n and\u00a0 \r\na<\/mi>5<\/mn><\/msub><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n