thomasthetankengine
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- Jun 19, 2014
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In this question, f and g are the functions given by
f(x)=ln(2−x) and g(x)=ln(6-x).
By writing ln(2−x)=ln2+ln(1-1/2x) and using substitution in one of the standard Taylor series given in the module, find the Taylor series about 0 for f. Give explicitly all terms up to the term in x3. Determine a range of validity for this
Taylor series.
Use your answer to part (a), and the fact that 6 − x = 2 − (x − 4), to
find the Taylor series about 4 for g. Give explicitly the same number
of terms as in part (a). Determine a range of validity for this Taylor
series.
Check the first four terms in the Taylor series that you found in
part (b) by finding the first, second and third derivatives of g, and
using these to find the cubic Taylor polynomial about 4 for g.
f(x)=ln(2−x) and g(x)=ln(6-x).
By writing ln(2−x)=ln2+ln(1-1/2x) and using substitution in one of the standard Taylor series given in the module, find the Taylor series about 0 for f. Give explicitly all terms up to the term in x3. Determine a range of validity for this
Taylor series.
Use your answer to part (a), and the fact that 6 − x = 2 − (x − 4), to
find the Taylor series about 4 for g. Give explicitly the same number
of terms as in part (a). Determine a range of validity for this Taylor
series.
Check the first four terms in the Taylor series that you found in
part (b) by finding the first, second and third derivatives of g, and
using these to find the cubic Taylor polynomial about 4 for g.
