Challenging Maclaurin’s series problem 1

If Tex2Img_1402896309,

(i) find the Maclaurin’s series of f(x) up to and including the term in Tex2Img_1402896490.

(ii) Hence deduce the series expansion of the function Tex2Img_1402896783 up to and including the term in Tex2Img_1402896839

Solution

(i) is routine, can easily work out to be

Tex2Img_1402896636

(ii) requires Higher Order Thinking (HOT)

We differentiate the series obtained in (i) to get Tex2Img_1402897125

Tex2Img_1402897294

Principle used: Differentiate one series expansion to get the series expansion for another series.

Quick Maclaurin’s series expansion using GC

In Maclaurin’s series expansion, where up to f”(0) need to be evaluated, can be done quickly using the GC.

Example

Expand Tex2Img_1402541222 up to powers of 2.

Using TI 84 Plus (OS 2.55)

maclaurin

From G.C,

f(0)= 3/2

f'(0) = -3/4

f”(0)= 5/4

Therefore using Maclaurin’s series expansion,

Tex2Img_1402541502

If powers higher than 2 such as 3 are required, then do differentiation to find the first derivative. Then enter into Y1. Y2 and Y3 will then give the second and third derivative.

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