Question

Use a 3rd degree Taylor polynomial centered at x = 1 for f(x) = ln(x) to approximate ln(1.2) to 5 decimal places.

Thank you!!

Answer 1

Compute the first 3 derivatives of f(x):

`f(x)=\ln x\implies f'(x)=\frac1x\implies f''(x)=-\frac1{x^2}\implies f'''(x)=\frac2{x^3}`

The 3rd degree Taylor polynomial about
`x=1` is then

`T_3(x)=f(1)+(f'(1))/(1!)(x-1)+(f''(1))/(2!)(x-1)^2+(f'''(1))/(3!)(x-1)^3`

`T_3(x)=(x-1)-(x-1)^2+2(x-1)^3`

`T_3(x)=-4+9x-7x^2+2x^3`

Now use
`T_3` to approximate ln(1.2):

`f(x)\approx T_3(x)\implies \ln1.2\approx T_3(1.2)=\boxed{0.17600}`

(Compare to the actual value, closer to 0.18232)