Solved Problems: Average Rate of Change of Functions with Detailed Steps

📚 Understanding Average Rate of Change

The average rate of change measures how much a function's output changes per unit change in its input over a specific interval. It's essentially the slope of the secant line connecting two points on the function's graph.

📜 Historical Context

The concept of average rate of change has roots in calculus and the development of understanding motion and change. Early mathematicians like Newton and Leibniz explored these ideas, laying the groundwork for calculus as we know it today.

📌 Key Principles

• 📈 Definition: The average rate of change of a function $f(x)$ over the interval $[a, b]$ is given by the formula: $\frac{f(b) - f(a)}{b - a}$.
• 📐 Geometric Interpretation: It represents the slope of the secant line that passes through the points $(a, f(a))$ and $(b, f(b))$ on the graph of $f(x)$.
• ⏱️ Units: The units of the average rate of change are the units of $f(x)$ per unit of $x$.

➗ Solved Problems: Detailed Steps

Problem 1

Find the average rate of change of $f(x) = x^2 + 2x - 3$ over the interval $[1, 4]$.

1. 🔢 Step 1: Calculate $f(4)$. $f(4) = (4)^2 + 2(4) - 3 = 16 + 8 - 3 = 21$.
2. 🔢 Step 2: Calculate $f(1)$. $f(1) = (1)^2 + 2(1) - 3 = 1 + 2 - 3 = 0$.
3. ➗ Step 3: Apply the formula. Average rate of change $= \frac{f(4) - f(1)}{4 - 1} = \frac{21 - 0}{4 - 1} = \frac{21}{3} = 7$.

Problem 2

Find the average rate of change of $g(x) = \sqrt{x}$ over the interval $[4, 9]$.

1. 🔢 Step 1: Calculate $g(9)$. $g(9) = \sqrt{9} = 3$.
2. 🔢 Step 2: Calculate $g(4)$. $g(4) = \sqrt{4} = 2$.
3. ➗ Step 3: Apply the formula. Average rate of change $= \frac{g(9) - g(4)}{9 - 4} = \frac{3 - 2}{9 - 4} = \frac{1}{5}$.

Problem 3

Find the average rate of change of $h(x) = \frac{1}{x}$ over the interval $[2, 6]$.

1. 🔢 Step 1: Calculate $h(6)$. $h(6) = \frac{1}{6}$.
2. 🔢 Step 2: Calculate $h(2)$. $h(2) = \frac{1}{2}$.
3. ➗ Step 3: Apply the formula. Average rate of change $= \frac{h(6) - h(2)}{6 - 2} = \frac{\frac{1}{6} - \frac{1}{2}}{6 - 2} = \frac{\frac{1 - 3}{6}}{4} = \frac{-\frac{2}{6}}{4} = -\frac{1}{12}$.

Problem 4

The height of a ball thrown upwards is given by $s(t) = -5t^2 + 30t + 2$, where $t$ is in seconds and $s(t)$ is in meters. Find the average velocity (rate of change) from $t = 1$ to $t = 3$ seconds.

1. ⏱️ Step 1: Calculate $s(3)$. $s(3) = -5(3)^2 + 30(3) + 2 = -45 + 90 + 2 = 47$.
2. ⏱️ Step 2: Calculate $s(1)$. $s(1) = -5(1)^2 + 30(1) + 2 = -5 + 30 + 2 = 27$.
3. ➗ Step 3: Apply the formula. Average velocity $= \frac{s(3) - s(1)}{3 - 1} = \frac{47 - 27}{3 - 1} = \frac{20}{2} = 10$ m/s.

Problem 5

Find the average rate of change of $f(x) = e^x$ over the interval $[0, 1]$.

1. 🔢 Step 1: Calculate $f(1)$. $f(1) = e^1 = e$.
2. 🔢 Step 2: Calculate $f(0)$. $f(0) = e^0 = 1$.
3. ➗ Step 3: Apply the formula. Average rate of change $= \frac{f(1) - f(0)}{1 - 0} = \frac{e - 1}{1} = e - 1 \approx 1.718$.

Problem 6

The temperature $T$ (in degrees Celsius) of a chemical reaction $t$ minutes after the reaction started is given by $T(t) = 3t^2 - t + 5$. Find the average rate of change of temperature during the first 2 minutes.

1. 🌡️ Step 1: Calculate $T(2)$. $T(2) = 3(2)^2 - 2 + 5 = 12 - 2 + 5 = 15$.
2. 🌡️ Step 2: Calculate $T(0)$. $T(0) = 3(0)^2 - 0 + 5 = 0 - 0 + 5 = 5$.
3. ➗ Step 3: Apply the formula. Average rate of change $= \frac{T(2) - T(0)}{2 - 0} = \frac{15 - 5}{2} = \frac{10}{2} = 5$ °C/min.

Problem 7

Find the average rate of change of $f(x) = \ln(x)$ over the interval $[1, e]$.

1. 🔢 Step 1: Calculate $f(e)$. $f(e) = \ln(e) = 1$.
2. 🔢 Step 2: Calculate $f(1)$. $f(1) = \ln(1) = 0$.
3. ➗ Step 3: Apply the formula. Average rate of change $= \frac{f(e) - f(1)}{e - 1} = \frac{1 - 0}{e - 1} = \frac{1}{e - 1} \approx 0.582$.

🌍 Real-World Examples

• 🚗 Driving Speed: Calculating the average speed of a car over a certain distance.
• 🌱 Population Growth: Determining the average growth rate of a population over a period of time.
• 🌡️ Temperature Change: Measuring the average change in temperature over a day.

💡 Conclusion

The average rate of change is a fundamental concept with broad applications. By understanding its definition and geometric interpretation, you can solve a variety of problems across different fields. Practice is key to mastering this concept!