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.
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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).]](https://www.doubtrix.com/js/ckeditor/filemanager/connectors/php/editor/1702721331-Bbgfyyfjfyfigh.png)
