I have a question about using the linearization of a function to estimate a given value.
Let $f(x)=\ (5x+3)^{\frac{1}{3}}+\sin(\frac{\pi}{2}x)+\ln(2x-1)$
$(a)$ Find the linearization of $f$ at $a=1$.
$(b)$ Use the linear approximation to evaluate $(7.5)^{\frac{1}{3}}+\sin(\frac{9\pi}{20})+\ln(\frac{4}{5})$
Part $(a)$ was easy. I got $L(x)=\frac{29}{12}x+\frac{7}{12}$
How do I go about doing part b though? Would I just plug in $x=0.90$? Since that is the point where $5x+3=7.5,\frac{\pi}{2}x=\frac{9\pi}{20},2x-1=\frac{4}{5}$. I am not sure though. Can someone advice me if I am going about this the right way?
Thanks!