Question

find the average rate of change of the function over the given interval. r(θ) = √(4θ + 1); 2,12 δr/δθ = (simplify your answer.)

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( y = f(x) \) over the interval \([a, b]\) is given by \(\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b - a}\). For the function \( R(\theta)=\sqrt{4\theta + 1} \) and the interval \([2,12]\), we have \( a = 2 \), \( b=12 \), so we need to find \( R(12) \) and \( R(2) \) first.

Step2: Calculate \( R(12) \)

Substitute \( \theta=12 \) into \( R(\theta) \):
\( R(12)=\sqrt{4\times12 + 1}=\sqrt{48 + 1}=\sqrt{49} = 7 \)

Step3: Calculate \( R(2) \)

Substitute \( \theta = 2 \) into \( R(\theta) \):
\( R(2)=\sqrt{4\times2+1}=\sqrt{8 + 1}=\sqrt{9}=3 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{\Delta R}{\Delta\theta}=\frac{R(12)-R(2)}{12 - 2}\), substitute the values of \( R(12) \) and \( R(2) \):
\(\frac{\Delta R}{\Delta\theta}=\frac{7 - 3}{12 - 2}=\frac{4}{10}=\frac{2}{5}\)

Answer:

\(\frac{2}{5}\)