I need a help to solve these four calculus questions.

1.

(2 marks) Consider the Maclaurin series for e" and cosh x : 2.2k and cosh x = K! k= (2k) ! (i) Using the power series above, it follows that the Maclaurin series for ed is given by 34 = A + Ba + Caz + Das + ... where A = Q P , B = C = and D = P (ii) Using the power series above, or otherwise, calculate the Taylor polynomial of degree 3 about 0 for est cosh x . P3(2) = P(1 mark) Consider the Maclaurin series for sin x and cos x OO sin x = C(-1)k x2k+1 and COSC= E(-1)k 2 2k k=0 (2k + 1)! k=0 (2k)! valid for all r E R . Using the power series above and the identity sin(x + a) = sin x cos a + cos x sin a, it follows that the Maclaurin series for sin (a + - is given by sin (a + Chack where the exact values of c4, C7 are CA = and C7 =(2 marks) Consider the Maclaurin series for sin x and cos x Sinx = (-1)4 22k+] and Cosa = >(-1)2 22% K=0 (2k + 1)! K=0 (2k)! Using the power series above it follows that the Maclaurin series for cosa is given by OO 2 2k cos x = A+ Ba + Cat...= At C (-1)ak (2k)! k=1 where A = SIP, B = GOD, C= P and ak (Hint : use the trigonometric identity for cos (a) in terms of cos(2ac) )( 2 marks ) Consider the Maclaurin series for 1 1 - x 1 - x x E (-1, 1) . (i) Using the power series above, it follows that the Maclaurin series for 1 is given by 1 - 1 M8 aback x E ( - 7, 7) 1 - ~/8 K=0 where ak = Now we can get 1 black 7 - 20 x E ( - 7, 7) K=0 where bk = 1 1 (ii) Using the power series above, or otherwise, calculate the Maclaurin series for (7 - 20) 2 7 - 2 7 - 2 (7 - 20) 2 = A + Ba + Ca2 + Das + ... . I E ( - 7, 7 ) where A = ADD , B= C = and D =