🎯 Understanding Intervals
What is an Interval?
An interval [a, b] represents all real numbers between and including points a and b on the number line.
📏 Types of Intervals
🔒 Closed Interval [a, b]
Notation: [a, b]
Meaning: Includes both endpoints a and b
Example: [2, 5] includes 2, 3, 4, and 5
Usage: Most common in rate of change problems
🔓 Open Interval (a, b)
Notation: (a, b)
Meaning: Excludes both endpoints a and b
Example: (2, 5) includes numbers between 2 and 5, but not 2 or 5
Usage: Less common in basic rate problems
🔐 Half-Open Intervals
Notation: [a, b) or (a, b]
Meaning: Includes one endpoint, excludes the other
Example: [2, 5) includes 2 but excludes 5
Usage: Special cases in advanced problems
🧮 The Interval Formula
📊 Step-by-Step Process
Example 1: Quadratic Function Over [1, 4]
Problem: Find the average rate of change of f(x) = x² - 3x + 2 over the interval [1, 4]
🔍 Interval Length Impact
Comparing Different Interval Lengths
Let's see how interval length affects the average rate of change for f(x) = x²:
| Interval | Length | f(a) | f(b) | Average Rate |
|---|---|---|---|---|
| [1, 2] | 1 | 1 | 4 | 3 |
| [1, 3] | 2 | 1 | 9 | 4 |
| [1, 4] | 3 | 1 | 16 | 5 |
| [1, 5] | 4 | 1 | 25 | 6 |
🌟 Real-World Interval Applications
📈 Business: Quarterly Revenue Analysis
Scenario: A company's revenue R(t) = 2t² + 5t + 10 thousand dollars, where t is months since January.
Problem: Find the average rate of revenue change over Q1 (months 0-3) and Q2 (months 3-6).
Q1 Analysis [0, 3]:
R(3) = 2(3)² + 5(3) + 10 = 18 + 15 + 10 = 43 thousand
R(0) = 2(0)² + 5(0) + 10 = 10 thousand
Q1 Average Rate = (43 - 10) / (3 - 0) = 33 / 3 = 11 thousand/month
Q2 Analysis [3, 6]:
R(6) = 2(6)² + 5(6) + 10 = 72 + 30 + 10 = 112 thousand
R(3) = 43 thousand (from above)
Q2 Average Rate = (112 - 43) / (6 - 3) = 69 / 3 = 23 thousand/month
🚗 Physics: Motion Analysis
Scenario: A car's position s(t) = t³ - 6t² + 9t meters at time t seconds.
Problem: Compare average velocity over intervals [0, 2], [2, 4], and [4, 6].
Interval [0, 2]:
s(2) = 8 - 24 + 18 = 2 meters
s(0) = 0 meters
Average velocity = (2 - 0) / (2 - 0) = 1 m/s
Interval [2, 4]:
s(4) = 64 - 96 + 36 = 4 meters
s(2) = 2 meters
Average velocity = (4 - 2) / (4 - 2) = 1 m/s
Interval [4, 6]:
s(6) = 216 - 216 + 54 = 54 meters
s(4) = 4 meters
Average velocity = (54 - 4) / (6 - 4) = 25 m/s
⚠️ Common Interval Mistakes
🚫 Mistake 1: Incorrect Interval Notation
Wrong: Using (a, b) when the problem specifies [a, b]
Impact: May exclude important boundary values
Solution: Always check if endpoints are included
🚫 Mistake 2: Swapped Endpoints
Wrong: Using [b, a] instead of [a, b] where a < b
Impact: Results in negative denominators and sign errors
Solution: Always ensure a < b for proper interval notation
🚫 Mistake 3: Misreading Interval Length
Wrong: Calculating (b - a) incorrectly
Impact: Incorrect final answer
Solution: Double-check arithmetic: larger number minus smaller number
🎯 Interval Selection Strategy
Choosing Appropriate Intervals
- Equal Length Intervals: Use for comparing rates across different periods
- Natural Boundaries: Use meaningful endpoints (quarters, years, etc.)
- Function Behavior: Consider where the function changes behavior
- Problem Context: Let the real-world situation guide interval choice
📚 Practice Problems
Problem 1: For f(x) = 2x³ - x² + 4, find the average rate of change over:
- a) [0, 1]
- b) [1, 2]
- c) [0, 2]
Problem 2: A population P(t) = 1000 + 50t + 2t² grows over time t years. Compare average growth rates over intervals [0, 5], [5, 10], and [10, 15].
Problem 3: Temperature T(h) = 70 - 3.5h°F at altitude h thousand feet. Find average temperature change over intervals [0, 2], [2, 4], and [4, 6].