Question

3. Answer the following questions. Justify your answers. A. (&pts) Find the first three nonzero terms of the Taylor series of the function f(x) = px? centered at a = -1. B. (&pts) What is the Maclaurin series for f(x) = sin x? Use it to obtain the Maclaurin series for the function g(x) = x. sin 2x. C. (4pts) Suppose that f(2)= 1, f'(2) = 3, and f" (2) = -2. How can you use this information to estimate the value of f(2.1)? Find the best estimate you can for f(2.1).

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Answer 1

Solution <A> Given f(x) = ep2 centered at aardie 20 = -1 2! (-1)2 f(-1)= e se. 2 X20 + The Taylor series is, f(2) = fline) + pop f'(m) + (x-70 ) {"(20)+ --- Here, f(200) f'(X) = arep 3x f'(-1) = 2 (-1)&no) -Re. f"cm) = 2 [axe**23 teaux ? = f'(*) >f(-1) = 4(-1)² e' tel = f'(-1) = 5e series written as, f(a) = (x+1)x-de + (a.tv) * se torno 4x²2x² tena can The Taylor et 2

f(x)=e slager 22.x + (22+1+2x) x2152t.. = f(a) serae aalonta 2.50x² + 2050 a seator + f(x) = (e-de +2.50) (5e-dela + 2.500?p.. = f(a) = 1.se + 3e.xt 2.50n ex²t we know, es go718 ☆ f(a) = 4.08 + 8.16x + 6.8x i fêrst three non-zero terms of f(x) are, f(x) = 4.08 + 8.168 +6.842. 2 f

+ ) sina LB We know, Taylor series, defined as F(x) = f(x0) + 7-20 fino) (0-20) ? = "(MD)+.. I! 2! At x= xoro then Taylor series is known as Maclaurin series. so Mac laurin series can written ans, FCx) = f(0) + 2 floo) + ma? F"(0) + na. Given f(a) so FCO) = 0, F COC] = cosa o fico)= 1. f'(X) = -fina a pll(o)=0 f!!!(m) - cosa > Full0)=-1. Therefore, FCM) = 0 +(ax 1 ) Hladna xo ) + max > Fla) = 3! 5! 7! . Mae laurin series for f(x) a sinn is fca) = 23 25 3! 5! 7! * Maclaurin series for the function. ger) =x.binaa sax F (2x) x 1t. 5 2-43 x x 7 + + 7 2. + 7 a +

tycase of un- maybe some for you (2a- > gen) = X 23 3225 12827 of + 82 31 .. ) 5! 7! g(x) = ax? ega? 8x4 3! 32a 5! 12828 + + 7! - We know, 2! Ley fiven, f(2)= 1, f'(2) = 3, f'(22=-2, F(x) = f(x0) + 270 f'(25) + (2-20 )? ["(20)+ -as + F() = f(a) + (-a) F'(a) + (2-2)2 4 "(2) + .. = F() = = 1 +(2-2) x 3 x²+4-48 At 20=2. 3 + 1 - 2 t -- 2 above series s) Fla) = 1 + 31-6=x2-4 +4x + -42+72-9 +--- Let's take the series up to 3 - terms, F(2) -42 +7-9. (201)2+ +7(2:1)-9 = F(2:1) 4:41 + 14.7-9 = F (2:1) = 1.29 = f(2:1) =