3.3-4. 4. Find the 3rd order approximation polynomial at the origin of the following function.

0|7, 2 3 10Hol| 9/8H e* sin x = x+ x2 +- (1) f(x) = e" sina (2) f(x) = x log(1 + x) 31 (3) f(x) = T - cos x (4) f(a) = 2" Avlcto talo (5) f(x) = VI+x (6) f(x) = (1+2)-1 ed sinx - (x+x2+ 2x3) =x4(... ) =0(x3) (7) f() = ( +1)(2+2) (8) 1(2) =72 +2+2 3 ! i) f(0), f'(0), ..., f(") (0)2 25 49 78/LE, ex = 1+ x+ + + o(x3 ) JEr BIG LICH. sin x = x - + o(x3) (1) ii) 9 52 0186H ExC. = e* sinx = x+x2+- 3 ! "+ 0(x3 ) x 2 ex = 1+x+- 2! + ..., sin x = x - 3 ! - + ... 73 (2) 73(x) =- 1+x+ (8) 1 (3) 13(x) = -1+x+ x2 + 2x+2 2 1+ (x+ x2/2) x' (log 2)3 0|7, - 1 1 (4) 73(x) = 1 + xlog 2+ x(log 2)2 2 - (1 - 1+12 - 13 ) + 0(13 )0|BE, + . 2 2 6 1 (5) 73 ( x) = 1 + - - x2 + 2x + 2 - (1 - (x+x2/2) + (x+ x2/2)2 - (x+x2/2)3) + 0( ( x + x 2 /2 ) 2 8 + 16 1 (6) 13(x) = 1-x+x2-x3 2 2 4 - + 0(x3 ) 1 3 OICH (7) 13(x) =- -x+- 1 2_ 15 2 4" 8" 16 x2 + 2x+ 2 1+ (x + 1)2- 1 +t -= 1-1+12 -13 + 0(13 ) 28 0186HE(1) f(x) = e" sin x (2) f(x) = xlog(1 + x) (3) f(x) = x- Cos (4) f(20) = 20 (5) f(ac) = V1+ x (6) f(ac) = (1 + 2) -1 1 1 (7 f(x) = 8 f(x) = ( ac + 1) (2 + 2 ) ac2 + 2x + 2