Use Taylor’s formula with a = 0 and n = 3 to find the standard cubic approximation of f(x) = 1/1-x at x = 0. Give an upper bound for the magnitude of the error in the approximation when |x| ≤ 0.1. What is the standard cubic approximation? P3(x) = According to the Remainder Theorem, what is the smallest upper bound for the error of this approximation when |x| ≤ 0.1 ? The upper bound is using the smallest possible value for M. (Use scientific notation. Round to two decimal places as needed. Use the multiplication symbol in the math palette as needed.)
Consider the function f(x) = sin(x)/x. 1. Compute limx→0 f(x) using l’Hôpital’s rule. 2. Use Taylor’s remainder theorem to get the same result: (a) Write down P1(x), the first-order Taylor polynomial for sin(x) centered at a = 0. (b) Write down an upper bound on the absolute value of the remainder R1(x) = sin(x) - P1(x), using your knowledge about the derivatives of sin(x). (c) Express f(x) as f(x) = P1(x)/x + R1(x)/x, and compute the limits of the two terms as x → 0.
Let f(x) = x^-1/2. The first four derivatives of f(x) are f’(x) = -1/2x^-3/2, f’’(x) = 3/4 x^-5/2, f’’’(x) = -15/8x^-7/2, fiv(x) = 105/16x^-9/2. (a) Write down T3, the Taylor polynomial of degree 3 for f(x) centred at 4 . (b) Use Taylor's remainder theorem (Property 2.9) to give an upper bound on the error when using T3 to estimate 1/√2. [Hint: for part (b) you need to write down the formula for the error E3(x) and find an upper bound for this on the interval (2,4).]