Calculus and Above

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In this question, f and...

In this question, f and g are functions given by:

f(x) = ln(2-x) and g(x) = ln(6-x)

a) By writing ln(2-x)=ln2+ln(1-x/2), and using substitution in one of the standard Taylor series, fine the Taylor series about 0 for f. Give explicitly all terms up to the term in x^3. Determine a range of validity for this Taylor series.

b) 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.

c) 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)

a) By writing ln(2-x)=ln2+ln(1-x/2), and using substitution in one of the standard Taylor series, fine the Taylor series about 0 for f. Give explicitly all terms up to the term in x^3. Determine a range of validity for this Taylor series.

b) 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.

c) 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.

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