Mathematics

Problem

Let

f(x)=\dfrac{\ln(1+3x)}{x}

for

x\neq0

and

f(0)=\displaystyle \lim_{x\to0}\dfrac{\ln(1+3x)}{x}

. a. Determine the first three non-zero terms of the Maclaurin series for

f(x)

and write them in ascending powers of

x

. Show all reasoning. b. Use your series from part (a) to evaluate

f(0)

and verify that the limit defining

f(0)

exists. c. Using the third-degree Maclaurin polynomial, approximate

f(0.05)

. Give your result to five decimal places. d. State whether the approximation in part (c) is an overestimate or an underestimate of the exact value of

f(0.05)

. Justify your answer analytically.

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