Calculate Average Rate of Change - Examples & Practice

📐 Remember: Average Rate of Change = (f(b) - f(a)) / (b - a)

🌱 Beginner

🌿 Intermediate

🌳 Advanced

🚀 Applications

🌱 Beginner Examples

Example 1: Linear Function

Problem: Find the average rate of change of f(x) = 2x + 3 from x = 1 to x = 5

Step 1: Identify interval [a, b] = [1, 5]

Step 2: Calculate f(5) = 2(5) + 3 = 13

Step 3: Calculate f(1) = 2(1) + 3 = 5

Step 4: Apply formula: (13 - 5) / (5 - 1) = 8 / 4 = 2

Answer: 2 units per unit

💡 For linear functions, the average rate of change equals the slope!

Example 2: Coordinate Points

Problem: Find the average rate of change between points (2, 7) and (6, 19)

Step 1: Identify coordinates: (x₁, y₁) = (2, 7), (x₂, y₂) = (6, 19)

Step 2: Calculate Δy = 19 - 7 = 12

Step 3: Calculate Δx = 6 - 2 = 4

Step 4: Apply formula: 12 / 4 = 3

Answer: 3 units per unit

Example 3: Simple Quadratic

Problem: Find the average rate of change of f(x) = x² from x = 0 to x = 3

Step 1: Interval [0, 3]

Step 2: f(3) = 3² = 9

Step 3: f(0) = 0² = 0

Step 4: (9 - 0) / (3 - 0) = 9 / 3 = 3

Answer: 3 units per unit

Example 4: Negative Values

Problem: Find the average rate of change of f(x) = -x + 4 from x = 1 to x = 4

Step 1: Interval [1, 4]

Step 2: f(4) = -4 + 4 = 0

Step 3: f(1) = -1 + 4 = 3

Step 4: (0 - 3) / (4 - 1) = -3 / 3 = -1

Answer: -1 units per unit (decreasing)

🌿 Intermediate Examples

Example 5: Complex Quadratic

Problem: Find the average rate of change of f(x) = 2x² - 3x + 1 from x = -1 to x = 3

Step 1: Interval [-1, 3]

Step 2: f(3) = 2(9) - 3(3) + 1 = 18 - 9 + 1 = 10

Step 3: f(-1) = 2(1) - 3(-1) + 1 = 2 + 3 + 1 = 6

Step 4: (10 - 6) / (3 - (-1)) = 4 / 4 = 1

Answer: 1 unit per unit

Example 6: Cubic Function

Problem: Find the average rate of change of f(x) = x³ - 2x² + x from x = 0 to x = 2

Step 1: Interval [0, 2]

Step 2: f(2) = 8 - 8 + 2 = 2

Step 3: f(0) = 0 - 0 + 0 = 0

Step 4: (2 - 0) / (2 - 0) = 2 / 2 = 1

Answer: 1 unit per unit

Example 7: Rational Function

Problem: Find the average rate of change of f(x) = 1/x from x = 1 to x = 4

Step 1: Interval [1, 4]

Step 2: f(4) = 1/4 = 0.25

Step 3: f(1) = 1/1 = 1

Step 4: (0.25 - 1) / (4 - 1) = -0.75 / 3 = -0.25

Answer: -0.25 units per unit

Example 8: Square Root Function

Problem: Find the average rate of change of f(x) = √x from x = 1 to x = 9

Step 1: Interval [1, 9]

Step 2: f(9) = √9 = 3

Step 3: f(1) = √1 = 1

Step 4: (3 - 1) / (9 - 1) = 2 / 8 = 0.25

Answer: 0.25 units per unit

🌳 Advanced Examples

Example 9: Exponential Function

Problem: Find the average rate of change of f(x) = 2^x from x = 1 to x = 4

Step 1: Interval [1, 4]

Step 2: f(4) = 2⁴ = 16

Step 3: f(1) = 2¹ = 2

Step 4: (16 - 2) / (4 - 1) = 14 / 3 ≈ 4.67

Answer: 14/3 ≈ 4.67 units per unit

Example 10: Trigonometric Function

Problem: Find the average rate of change of f(x) = sin(x) from x = 0 to x = π/2

Step 1: Interval [0, π/2]

Step 2: f(π/2) = sin(π/2) = 1

Step 3: f(0) = sin(0) = 0

Step 4: (1 - 0) / (π/2 - 0) = 1 / (π/2) = 2/π ≈ 0.637

Answer: 2/π ≈ 0.637 units per unit

Example 11: Logarithmic Function

Problem: Find the average rate of change of f(x) = ln(x) from x = 1 to x = e

Step 1: Interval [1, e]

Step 2: f(e) = ln(e) = 1

Step 3: f(1) = ln(1) = 0

Step 4: (1 - 0) / (e - 1) = 1 / (e - 1) ≈ 0.582

Answer: 1/(e-1) ≈ 0.582 units per unit

Example 12: Piecewise Function

Problem: For f(x) = {x² if x ≤ 2; 2x if x > 2}, find average rate from x = 1 to x = 3

Step 1: Interval [1, 3]

Step 2: f(3) = 2(3) = 6 (since 3 > 2)

Step 3: f(1) = 1² = 1 (since 1 ≤ 2)

Step 4: (6 - 1) / (3 - 1) = 5 / 2 = 2.5

Answer: 2.5 units per unit

🚀 Real-World Applications

Physics: Projectile Motion

Problem: A ball's height h(t) = -16t² + 64t + 5 feet. Find average velocity from t = 1 to t = 3 seconds.

Step 1: Interval [1, 3]

Step 2: h(3) = -16(9) + 64(3) + 5 = -144 + 192 + 5 = 53 ft

Step 3: h(1) = -16(1) + 64(1) + 5 = -16 + 64 + 5 = 53 ft

Step 4: (53 - 53) / (3 - 1) = 0 / 2 = 0 ft/s

Answer: 0 ft/s (ball returns to same height)

Economics: Profit Analysis

Problem: Company profit P(x) = -x² + 12x - 20 thousand dollars for x units. Find average rate from 2 to 6 units.

Step 1: Interval [2, 6]

Step 2: P(6) = -36 + 72 - 20 = 16 thousand

Step 3: P(2) = -4 + 24 - 20 = 0 thousand

Step 4: (16 - 0) / (6 - 2) = 16 / 4 = 4 thousand/unit

Answer: $4,000 profit per unit increase

Biology: Population Growth

Problem: Bacteria population N(t) = 100e^(0.2t) after t hours. Find average growth rate from t = 0 to t = 5.

Step 1: Interval [0, 5]

Step 2: N(5) = 100e^(1) = 100e ≈ 271.8

Step 3: N(0) = 100e^0 = 100

Step 4: (271.8 - 100) / (5 - 0) = 171.8 / 5 ≈ 34.4

Answer: ≈34.4 bacteria per hour

Chemistry: Reaction Rate

Problem: Concentration C(t) = 0.5e^(-0.1t) mol/L after t minutes. Find average rate from t = 0 to t = 10.

Step 1: Interval [0, 10]

Step 2: C(10) = 0.5e^(-1) = 0.5/e ≈ 0.184 mol/L

Step 3: C(0) = 0.5e^0 = 0.5 mol/L

Step 4: (0.184 - 0.5) / (10 - 0) = -0.316 / 10 = -0.0316

Answer: -0.0316 mol/L per minute (decreasing)

🧮 Practice with Our Calculator

📝 Practice Problems

Try these problems on your own, then check with our calculator:

Problem Set A (Beginner)

1. f(x) = 3x - 1 from x = 2 to x = 5

2. Points (1, 4) and (7, 22)

3. f(x) = x² + 2 from x = 0 to x = 4

4. f(x) = -2x + 8 from x = 1 to x = 3

Problem Set B (Intermediate)

1. f(x) = x³ - 3x from x = -1 to x = 2

2. f(x) = 2/x from x = 1 to x = 5

3. f(x) = √(x + 1) from x = 0 to x = 8

4. f(x) = x² - 4x + 3 from x = 1 to x = 4

Problem Set C (Advanced)

1. f(x) = e^x from x = 0 to x = 2

2. f(x) = cos(x) from x = 0 to x = π

3. f(x) = ln(x + 1) from x = 0 to x = 3

4. f(x) = x^(3/2) from x = 1 to x = 9

📐 Formula Guide 📚 How to Calculate 📏 Interval Analysis 🔬 Calculus Applications