Let $f(x)=\sin x$. Find the exact value of the following expression. Do not use a calculator.
The average rate of change of from $x_{1}=\frac{5 \pi}{4}$ to $x_{2}=\frac{3 \pi}{2}$
Answer
Final Answer: The exact value of the average rate of change of the function \(f(x)=\sin x\) from \(x_{1}=\frac{5 \pi}{4}\) to \(x_{2}=\frac{3 \pi}{2}\) is \(\boxed{\frac{4(-1 + \sqrt{2}/2)}{\pi}}\).
Step 1 ：Given the function \(f(x)=\sin x\), we are asked to find the average rate of change from \(x_{1}=\frac{5 \pi}{4}\) to \(x_{2}=\frac{3 \pi}{2}\).
Step 2 ：The average rate of change of a function \(f(x)\) from \(x_{1}\) to \(x_{2}\) is given by the formula \(\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\).
Step 3 ：Substitute \(x_{1}=\frac{5 \pi}{4}\) and \(x_{2}=\frac{3 \pi}{2}\) into the function \(f(x)=\sin x\) to get \(f(x_{1})=-\sqrt{2}/2\) and \(f(x_{2})=-1\).
Step 4 ：Substitute these values into the formula for the average rate of change to get \(\frac{-1 - (-\sqrt{2}/2)}{\frac{3 \pi}{2} - \frac{5 \pi}{4}} = \frac{4(-1 + \sqrt{2}/2)}{\pi}\).
Step 5 ：Final Answer: The exact value of the average rate of change of the function \(f(x)=\sin x\) from \(x_{1}=\frac{5 \pi}{4}\) to \(x_{2}=\frac{3 \pi}{2}\) is \(\boxed{\frac{4(-1 + \sqrt{2}/2)}{\pi}}\).
