Linear Algebra with Applications, 10th Edition

5.7

1. 1. 

   1. (a) $T_4=8x^4-8x^2+1,T_5=16x^5-20x^3+5x$;

   2. (b) $\begin{array}{c}{H}_4=16x^4-48x^2+12,\hfill \\ H_5=32x^5-160x^3+120x\hfill \end{array}$

2. 2. $p_1(x)=x,p_2(x)=x^2-\frac{4}{\pi }+1$

3. 3. Hint: Let $x=\text{cos}\theta$ and make use of the identity $T_n\left(x\right)=\text{cos}n\theta$. This hint also applies to exercises 6, 7, and 8.

4. 4. $\begin{array}{l}p(x)=(\mathrm{sinh }1)P_0(x)+\frac{3}{e}{P}_1(x)+5(\mathrm{sinh }1-\frac{3}{e})P_2(x)\\ p(x)\approx 0.9963+1.1036x+0.5367x^2\end{array}$

5. 5. Hint: Write $p_n\left(x\right)$ in the form $p_n\left(x\right)=a_n{x}^n+q\left(x\right)$ where degree $q\left(x\right)<n$. By Theorem 5.7.1, $\langle q,p_n\rangle =0$. It follows then that

   ${\Vert p_n\Vert }^2=\langle p_n,p_n\rangle =\langle a_n{x}^n+q\left(x\right),p\left(x\right)\rangle$

6. 6. 

   1. (a) $U_0=1,U_1=2x{U}_2=4x^2-1$

7. 11. $p(x)=(x-2)(x-3)+(x-1)(x-3)+2(x-2)(x-2)$

8. 13. $1\cdot f(-\frac{1}{\sqrt{3}})+1\cdot f\left(\frac{1}{\sqrt{3}}\right)$

9. 14. (a) degree 3 or less; (b) the formula gives the exact answer for the first integral. The approximate value for the second integral is 1.5, while the exact answer is ${\displaystyle \frac{\pi }{2}}$.

10. 15.  

    1. (a) Hint: If f(x) is a ...