Tutorial Exercise Use the definition of Taylor series to find the Taylor series (centered at c) for the function. f(x) =

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Tutorial Exercise Use The Definition Of Taylor Series To Find The Taylor Series Centered At C For The Function F X 1 (30.33 KiB) Viewed 53 times

Tutorial Exercise Use The Definition Of Taylor Series To Find The Taylor Series Centered At C For The Function F X 2 (24.46 KiB) Viewed 53 times

Tutorial Exercise Use the definition of Taylor series to find the Taylor series (centered at c) for the function. f(x) = cos x, c = 4 Step 1 If a function f has derivatives of all orders at x = c, then the series (c) = f(c) + f '(c)(x = c) +- n f(c)x³ f"(c)(x-c)² 2! 31 n = 01 is called the Taylor series for f(x) centered at c. -cos(x) First we need to find the sequence of all derivatives for f(x). = COS X f(x) f'(x) = -sin x F"(x) = f"""(x) = sin(x) (4)(x) = cos(x) + -cos(x) sin (r) cos (1) +L+ (c)X + L nl

Step 2 Next, evaluate the derivatives at the center x= 4 Step 3 f(x) = cos x → (=/4) F'(x) = -sin x f'(x/4) F"(x) = -cos xf"(x/4) - Mix) BẢNK mà mơ (14) (4)(x) = cos x (4) (x/4) - C05 X = Σ A=0 √2 2 x 2 > > > Substitute the derivative values in the Taylor series centered at x-. 4 + and simplify. (Hint:Note that (-1) 13/2-1,-1,-1, 1, 1,-1,-1, 1) - (21/4) - [2-2-2-2 41 NS NS √2 (x-(x/4))² √2 (x-(x/4))³√2 (x-(n/4))4 + 21 2 31 Qaz 4 [MG-UY 2 By taytur cen