Module 1: Now You Try II

1. If $y = \cos (7 x)$ $y=\cos(7x)$ fill in the blanks for $y^{(n)}$ $y^{(n)}$ .

$y^{(n)} = {\begin{array}{lcccl} a^{n} \cos (\underline{}) & If n = 4 k \\ \underline{} \sin (\underline{}) & If n = 4 k + 1 \\ a^{n} \underline{} & If n = 4 k + 2 \\ \underline{} & If n = 4 k + 3 \end{array}$ $y^{(n)} \; =\; \left\{ \begin{array}{lcccl} a^n\cos(\underline{\hspace{.5cm}}) && &&\mbox{If}\;\;n=4k\\\\ \underline{\hspace{.5cm}}\sin(\underline{\hspace{.5cm}})&& &&\mbox{If}\;\;n=4k+1\\\\ a^n\underline{\hspace{.5cm}} &&&& \mbox{If}\;\;n=4k+2\\\\ \underline{\hspace{1cm}}&&&&\mbox{If}\;\; n=4k+3\\\\ \end{array} \right.$

Solution:

▶ Show

y^{(n)} = {\begin{array}{rcccl} a^{n} \cos (7 x) & If n = 4 k \\ - a^{n} \sin (7 x) & If n = 4 k + 1 \\ - a^{n} \cos (7 x) & If n = 4 k + 2 \\ a^{n} \sin (7 x) & If n = 4 k + 3 \end{array}

$y^{(n)} \; =\; \left\{ \begin{array}{rcccl} a^n\cos({\color{red}{7x}}) && &&\mbox{If}\;\;n=4k\\\\ {\color{red}{-a^{n}}}\sin({\color{red}{7x}})&& &&\mbox{If}\;\;n=4k+1\\\\ {\color{red}-}a^n{\color{red}\cos(7x)} &&&&\mbox{If}\;\; n=4k+2\\\\ {\color{red}a^n\sin(7x)}&&&& \mbox{If}\;\;n=4k+3\\\\ \end{array} \right.$

2. Consider $y = \sin (b x)$ $y=\sin(bx)$ . Match the expressions on the first list with conditions on the second list write to compute $y^{(n)}$ $y^{(n)}$ , depending on $n$ $n$ .