Worksheet 3 Solutions

K 2 3 4 c use the rst four terms of the taylor series

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Unformatted text preview: + ··· 2! 3! 4! 5! 6! 7! (d) Use the fact that ex = ei(−ix) to obtain the Taylor series for ex . Solution: Just replace x with −ix in the formula: ∞ x e =e i(−ix) = k=0 i(−ix) k! k ∞ = k=0 (−i2 x)k = k! ∞ k=0 xk . k! 4 Therefore, we have the following Taylor series for ex : ∞ ex = k=0 xk x2 x3 x4 =1+x+ + + + ··· . k! 2! 3! 4! (c) Use the first four terms of the Taylor series to estimate e = e1 . What does your calculator say is the value of e? Solution: Using the first four terms we have: e1 ≈ 1 + 1 + 1 1 8 + = ≈ 2.6667. 2! 3! 3 If you ask your calculator for the value if e, you should get something like e ≈ 2.71828. If we use more terms, we will get a better approximation. For example, if we use the first eight terms, we have: e1 ≈ 1 + 1 + 1 1 1 1 1 685 1 +++++ = ≈ 2.71825. 2! 3! 4! 5! 6! 27! 252 x2 2! Just for fun, let’s take a look at the graphs. In these graphs, f (x) = 1 + x + 2 3 4 5 6 7 g (x) = 1 + x + x + x + x + x + x + x . 2! 3! 4! 5! 6! 7! + x3 3! 400 150 y x 300 100 100 y fx 5 50 x 200 50 5 y 5 y 5 100 200 gx and 5 To give you a better idea of how these three compare, let’s plot ex , f (x), and g (x) at the same time: 200 y x y gx y 150 fx 100 50 5 5 50 100 You can see that the polynomial with eight terms is much more accurate than the one with four terms....
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This document was uploaded on 03/13/2014 for the course MATH 121 at Seattle University.

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