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- Expand f(x) = x^2 , 0 < x < pi in a cosine series, a sine series and a Fourier series. plot the extensions of this periodic function.Let g(x) = (x−1)^3/ (x-2)^2 . What is the value of the fifty derivative of the function g at the point 1: g^50(1)? Hint: Find the Taylor series of g centered at 1.Find the Taylor series generated by f(x) = cos 4x at x = 0, using the definition. Thanks
- Consider f(x) = 6.5x^5 - 3.7x^3 + 7x - 1.6. Using the given function, caculate the Rm by making use of the first two terms of Taylor series expansion to approximate f(2) given f(0) = -1.6.Expand f(x) = In x in powers of (x-1) using Taylor's Series. Compute In 1.2. Step 1: Write the given function f (x) = Step 2: Get its derivative (at least until fourth derivative) f (x) = f"(x) = f"(x) = fV (x) = Step 3: Let a = 1 (From x-1, where a =1) f (1) = f' (1) = = (1) ייr fV (1) = Step 4: Recall Taylor's Series and substitute values of f(0), f (0), r'(0) .. f'(a)(x=a), f"(a)(x-a)² f"(a)(x=a)³ 1! f"(a)(x-a)" f(x) = f (a) + +...+ 2! 3! n ! f(x) = Step 5: Try to represent it in summation. f(x) = sin (x) = Eo() Step 6: Evaluate In 1.2. Using the series, let x= 1.2. Use your calculator for checking. f(1.2) = In (1.2) =Compute the limit of the nth term and use it to explain why the seriesis divergent.