Mathematics

Mathematics

Problem

Consider the function gg defined for x>0x>0 by g(x)=ln⁡xxg(x)=\dfrac{\ln x}{x}. The function is to be analyzed near x=1x=1. (a) Starting from a known Taylor series, write the first four nonzero terms of the Taylor series of ln⁡x\ln x about x=1x=1. Express each term in powers of (x−1)(x-1). (b) Use your result from part (a) to obtain the first four nonzero terms of the Taylor series of gg about x=1x=1. Simplify each coefficient. (c) From your series in part (b), determine the exact values of g′′(1)g''(1) and g(4)(1)g^{(4)}(1). Clearly indicate how each derivative is obtained from the series.
Intuitive explanation

The key is recognizing that g(x)=ln⁡xxg(x) = \frac{\ln x}{x} behaves nicely near x=1x=1 because both numerator and denominator have predictable values there. Use a Taylor series expansion around x=1x=1 to capture how gg changes locally—this transforms the tricky ratio into a polynomial-like expression you can analyze term by term. This approach reveals the function's slope, curvature, and higher-order behavior without computing derivatives repeatedly.

Intuitive explanation

The key is recognizing that g(x)=ln⁡xxg(x) = \frac{\ln x}{x} behaves nicely near x=1x=1 because both numerator and denominator have predictable values there. Use a Taylor series expansion around x=1x=1 to capture how gg changes locally—this transforms the tricky ratio into a polynomial-like expression you can analyze term by term. This approach reveals the function's slope, curvature, and higher-order behavior without computing derivatives repeatedly.

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