121616949-math.285 - 10.12 Additional exercises 271 EXAMPLE...

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10.12 Additional exercises 271 EXAMPLE 10.54 Find a polynomial approximation for e x near x = 2 accurate to ± 0 . 005. From Taylor’s theorem: e x = N X n =0 e 2 n ! ( x - 2) n + e z ( N + 1)! ( x - 2) N +1 , since f ( n ) ( x ) = e x for all n . We are interested in x near 2, and we need to keep | ( x - 2) N +1 | in check, so we may as well specify that | x - 2 | ≤ 1, so x [1 , 3]. Also fl fl fl fl e z ( N + 1)! fl fl fl fl e 3 ( N + 1)! , so we need to find an N that makes e 3 / ( N + 1)! 0 . 005. This time N = 5 makes e 3 / ( N + 1)! < 0 . 0015, so the approximating polynomial is e x = e 2 + e 2 ( x - 2) + e 2 2 ( x - 2) 2 + e 2 6 ( x - 2) 3 + e 2 24 ( x - 2) 4 + e 2 120 ( x - 2) 5 ± 0 . 0015 . This presents an additional problem for approximation, since we also need to approximate e 2 , and any approximation we use will increase the error, but we will not pursue this complication. Exercises 1. Find a polynomial approximation for cos x on [0 , π ], accurate to ± 10 - 3 2. How many terms of the series for ln x centered at 1 are required so that the guaranteed error
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Unformatted text preview: , 3 / 2] is at most 10-3 ? What if the interval is instead [1 , 3 / 2]? ⇒ 3. Find the first three nonzero terms in the Taylor series for tan x on [-π/ 4 , π/ 4], and compute the guaranteed error term as given by Taylor’s theorem. (You may want to use Maple or a similar aid.) ⇒ 4. Show that cos x is equal to its Taylor series for all x by showing that the limit of the error term is zero, as N approaches infinity. 5. Show that e x is equal to its Taylor series for all x by showing that the limit of the error term is zero, as N approaches infinity. 10.12 Additional exercises These problems require the techniques of this chapter, and are in no particular order. Some problems may be done in more than one way....
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