Lanjutan Materi Turunan Fungsi Trigonometri (Matematika Peminatan Kelas XII)

$\begin{array}{l}\\ 3.& \text{Diketahui suatu gelombang bergerak teratur}\\ & \text{sebagaimana gambar berikut}\end{array}$

$.\begin{array}{l}\\ & \text{Gelombang tersebut pada waktu}t\text{detik mengikuti}\\ & \text{rumus}y=f(t)=2\sin {\displaystyle \frac{1}{2}\pi t.\text{Dan diketahui pula cepat}}\\ & \text{rambat gelomnya dapat dinyatakan dalam}\\ & v_t=\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{f(t+h)-f(t)}{h}\text{dengan}h\ne 0,\text{tentukanlah}:}\\ & \text{a}.\text{posisi gelombang pada ketika}t=1,5\text{detik}\\ & \text{b}.\text{rumus cepat rambat gelombang pada saat}t\\ & \text{c}.\text{cepat rambat gelombang saat}t=2{\displaystyle \frac{1}{2}\text{detik}}\end{array}$

$.\begin{array}{rl}& \text{Jawab}:\\ & \begin{array}{l}\\ \text{a}.& \text{Posisi gelombang saat}t=1,5={\displaystyle \frac{3}{2}\text{detik}}\\ & f(1,5)=2\sin {\displaystyle \frac{1}{2}\pi \left(\frac{3}{2}\right)}\\ & f(1,5)=2\sin {\displaystyle \frac{3}{4}\pi =2\sin {135}^{\circ }=2\left({\displaystyle \frac{1}{2}\sqrt{2}}\right)=\sqrt{2}}\\ \text{b}.& \text{Cepat rambat gelombang saat}t\text{detik}\\ & v_t=\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{f(t+h)-f(t)}{h}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{2\sin {\displaystyle \frac{1}{2}\pi (t+h)-2\sin {\displaystyle \frac{1}{2}\pi t}}}{h}}\\ & =\underset{h\to 0}{\text{Lim}}{\displaystyle \frac{2.2\cos \left({\displaystyle \frac{1}{2}t+\frac{1}{4}h}\right).\sin {\displaystyle \frac{1}{4}\pi h}}{h}}\\ & =4\times {\displaystyle \frac{1}{4}\pi \times \cos {\displaystyle \frac{1}{2}\pi t}}\\ & =\pi \cos {\displaystyle \frac{1}{2}\pi t}\\ \text{c}.& \text{Cepat rambat saat}t=2{\displaystyle \frac{1}{2}=\frac{5}{2}\text{detik}}\\ & v_t=\pi \cos {\displaystyle \frac{1}{2}\pi t}\\ & =\pi \cos {\displaystyle \frac{1}{2}\pi \left({\displaystyle \frac{5}{2}}\right)}\\ & =\pi \cos {\displaystyle \frac{5}{4}\pi }\\ & =\pi (-{\displaystyle \frac{1}{2}\sqrt{2}})\\ & =-{\displaystyle \frac{1}{2}\pi \sqrt{2}}\end{array}\end{array}$