Tiêu đề FP2 Maclaurins Series Past Paper Questions.pdf ✅

FP2 Maclaurins Expansion 1. June 2010 qu.2 It is given that f(x) = tan–1(1 + x). (i) Find f(0) and f ′(0), and show that f ′′(0) = 2 1 − . [4] (ii) Hence find the Maclaurin series for f(x) up to and including the term in x 2 . [2] 2. June 2009 qu.3 (i) Given that f(x) = esin x , find f′(0) and f′′(0). [4] (ii) Hence find the first three terms of the Maclaurin series for f(x). [2] 3. Jan 2009 qu. 1 (i) Write down and simplify the first three terms of the Maclaurin series for e2x . [2] (ii) Hence show that the Maclaurin series for ln(e2x +e–2x ) begins ln a +bx2 , where a and b are constants to be found. [4] 4. June 2008 qu. 7 It is given that f (x) = tanh−1 . 2 1 , for 2 1  −>      + − x x x (i) Show that f ′ (x) = - , 21 1 + x and find f ′′ (x). [6] (ii) Show that the first three terms of the Maclaurin series for f (x) can be written as ln a + bx + cx 2 , for constants a, b and c to be found. [4] 5. Jan 2008 qu. 1 It is given that f(x) = ln(1 + cos x). (i) Find the exact values of f(0), f ́(0) and f ̋(0). [4] (ii) Hence find the first two non-zero terms of the Maclaurin series for f(x). [2] 6. June 2007 qu. 2 (i) Given that f(x) = sin 2x 4   π +     , show that f(x) = 2 2 1 (sin 2x + cos 2x) [2] (ii) Hence find the first four terms of the Maclaurin series for f(x). [You may use appropriate results given in the List of Formulae.] [3]