In exercises 1-10, find ${\displaystyle P_n(x,f\left(x\right))}$ for the specified function and ${\displaystyle n}$.

1. ${\displaystyle f\left(x\right)=xsin\left(x\right)}$; ${\displaystyle n=6}$
   
   ${\displaystyle P_5(x,sin\left(x\right))=x-\frac{x^3}{6}+\frac{x^5}{5!}}$, so  ${\displaystyle P_6(x,xsin\left(x\right))=x{P}_5(x,sin\left(x\right))=x(x-\frac{x^3}{6}+\frac{x^5}{5!})=x^2-\frac{x^4}{6}+\frac{x^6}{5!}}$.
2. ${\displaystyle f\left(x\right)=x^{2}cos\left(x\right)}$; ${\displaystyle n=6}$
3. ${\displaystyle f\left(x\right)=xsin\left(x\right)+cos\left(x\right)}$; ${\displaystyle n=6}$
   ${\displaystyle P_6(x,cos\left(x\right))=1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}}$
   So ${\displaystyle P_6(x,xsin\left(x\right)+cos\left(x\right))=P_6(x,xsin\left(x\right))+P_6(x,cos\left(x\right))=x^2-\frac{x^4}{6}+\frac{x^6}{5!}+1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}=...}$
   
4. ${\displaystyle f\left(x\right)=x{e}^x}$; ${\displaystyle n=6}$
5. ${\displaystyle f\left(x\right)=sin\left(x^3\right)}$; ${\displaystyle n=9}$
   ${\displaystyle P_9(x,sin\left(x^3\right))=P_3(x^3,sin\left(x\right))=\left(x^3\right)-\frac{{\left(x^3\right)}^3}{6}=x^3 -\frac{x^9}{6}}$
   
6. ${\displaystyle f\left(x\right)=\frac{1}{2-x}}$; ${\displaystyle n=4}$
7. ${\displaystyle f\left(x\right)=\frac{1}{1+x}}$; ${\displaystyle n=4}$
8. ${\displaystyle f\left(x\right)=\frac{1}{1+x^2}}$; ${\displaystyle n=8}$
   
9. ${\displaystyle f\left(x\right)=}$arctan${\displaystyle \left(x\right)}$; ${\displaystyle n=8}$
   Hint: Use arctan'${\displaystyle \left(x\right)=\frac{1}{1+x^2}}$ and the previous problem.
   
10. ${\displaystyle f\left(x\right)=\frac{x}{2-x}}$; ${\displaystyle n=4}$
11. ${\displaystyle f\left(x\right)=e^{-2x}}$ ; ${\displaystyle n=4}$