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18 Nov 2019
6. Maclaurin polynomial The process of differentiation can be continued, provided that the derivatives exist The third derivative, denoted f"(x) or f(z), is the derivative of f"(x). More generally, the nth derivative fn) (a) is the derivative of the (n - 1)st derivative. We often refer to f(r) as the zeroth derivative and f'(x) as the first derivative. In Leibniz's notation, we write Let f be a function such that the first n derivatives exist at r-r. We define T,(x) the n-th Taylor polynomial for f based at r-r, as follows: where n!-n (n -1)...3-2 1 (Note that the degree of T is less than or equal to , since fm (r) could be zero.) In the special case where r0, we have: This is often called the n-th Maclaurin polynomial of f. Answer the following questions (a) Calculate the first three derivatives of f(x)-sin x. Then find the pattern and determine the n-th Maclaurin polynomial of sin x. (b) Calculate the first three derivatives of f(x) cos x. Then find the pattern and determine the n-th Maclaurin polynomial of cos r.
6. Maclaurin polynomial The process of differentiation can be continued, provided that the derivatives exist The third derivative, denoted f"(x) or f(z), is the derivative of f"(x). More generally, the nth derivative fn) (a) is the derivative of the (n - 1)st derivative. We often refer to f(r) as the zeroth derivative and f'(x) as the first derivative. In Leibniz's notation, we write Let f be a function such that the first n derivatives exist at r-r. We define T,(x) the n-th Taylor polynomial for f based at r-r, as follows: where n!-n (n -1)...3-2 1 (Note that the degree of T is less than or equal to , since fm (r) could be zero.) In the special case where r0, we have: This is often called the n-th Maclaurin polynomial of f. Answer the following questions (a) Calculate the first three derivatives of f(x)-sin x. Then find the pattern and determine the n-th Maclaurin polynomial of sin x. (b) Calculate the first three derivatives of f(x) cos x. Then find the pattern and determine the n-th Maclaurin polynomial of cos r.
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Sixta KovacekLv2
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