Question

f(x)=2sinx - 1

1. using the graph calculate the average rate of change in the interval 40≤x≤50
2. using the graph calculate the instantaneous rate of change at the point where x = 45°

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 40$, $b = 50$, and $f(x)=2\sin x-1$. So we need to find $f(40)$ and $f(50)$.
$f(40)=2\sin(40)-1$ and $f(50)=2\sin(50)-1$.

Step2: Calculate the average rate of change

$\text{Average rate of change}=\frac{f(50)-f(40)}{50 - 40}=\frac{(2\sin(50)-1)-(2\sin(40)-1)}{10}=\frac{2\sin(50)-2\sin(40)}{10}=\frac{2(\sin(50)-\sin(40))}{10}=\frac{\sin(50)-\sin(40)}{5}$.
Using a calculator (assuming angles are in degrees), $\sin(50)\approx0.766$, $\sin(40)\approx0.643$.
$\text{Average rate of change}=\frac{0.766 - 0.643}{5}=\frac{0.123}{5}=0.0246$.

Step3: Recall instantaneous - rate - of change (derivative)

The derivative of $y = f(x)=2\sin x-1$ is $f^\prime(x)=2\cos x$.

Step4: Evaluate the derivative at $x = 45^{\circ}$

When $x = 45^{\circ}$, $\cos(45^{\circ})=\frac{\sqrt{2}}{2}$. So $f^\prime(45)=2\cos(45)=2\times\frac{\sqrt{2}}{2}=\sqrt{2}\approx1.414$.

Answer:

1. The average rate of change in the interval $40\leq x\leq50$ is approximately $0.0246$.
2. The instantaneous rate of change at $x = 45^{\circ}$ is approximately $1.414$.