A function f(x) satisfies the following conditions:

f(1) = 0, $~~~~~~~~$ f (n) (1) = n for n = 1, 2, and 3,
3x ≤ f (4) (x) ≤ 9x provided x ≥ 1.

(a) What is the Taylor polynomial p 3 (x) of degree 3 for f(x) about x = 1?
(b) What is the approximate value for f(3/2) supplied by (x)?
(c) Based on the bounds for f (4) (x) given above, what is the smallest interval that you can be sure
contains the value f(3/2)?
(d) Based on your answer to (c), what is the best approximation you can give for f(3/2)?

Question

A function  f(x)  satisfies the following conditions:

f(1) = 0,	$~~~~~~~~$  f (n) (1) = n  for  n = 1, 2, and 3,
	3x  ≤  f (4) (x)  ≤  9x  provided  x  ≥  1.

(a)	What is the Taylor polynomial p 3 (x)  of degree 3  for  f(x)  about  x = 1?
(b)	What is the approximate value for  f(3/2)  supplied by   (x)?
(c)	Based on the bounds for  f (4) (x)  given above, what is the smallest interval that you can be sure
	contains the value  f(3/2)?
(d)	Based on your answer to (c), what is the best approximation you can give for  f(3/2)?

Quizplus · Accepted Answer

The Answer of A function  f(x)  satisfies the following conditions:

f(1) = 0,	$~~~~~~~~$  f (n) (1) = n  for  n = 1, 2, and 3,
	3x  ≤  f (4) (x)  ≤  9x  provided  x  ≥  1.

(a)	What is the Taylor polynomial p 3 (x)  of degree 3  for  f(x)  about  x = 1?
(b)	What is the approximate value for  f(3/2)  supplied by   (x)?
(c)	Based on the bounds for  f (4) (x)  given above, what is the smallest interval that you can be sure
	contains the value  f(3/2)?
(d)	Based on your answer to (c), what is the best approximation you can give for  f(3/2)?

A Function F(x) Satisfies the Following Conditions: ~~~~~~~~

A Function F(x) Satisfies the Following Conditions:
$~~~~~~~~$