Mathematics

Problem

Let

P(x)=a(x-3)^{2}+b(x-3)+c

be a quadratic polynomial chosen to approximate

\ln x

on an interval that includes

x=3

. The polynomial must satisfy

P(3)=\ln 3

P'(3)=\dfrac{1}{3}

, and

P(4)=\ln 4

. (a) Set up the system of equations implied by the three stated conditions and solve for

a

b

, and

c

. (b) Use the polynomial you found in part (a) to estimate

\ln(3.2)

. Give your answer correct to four decimal places.