The average rate of change is the slope of the secant line between two points on a function. It represents how much the function value changes per unit change in the input variable.
Imagine drawing a straight line connecting two points on a curve. The slope of this line is the average rate of change between those points.
For f(x) = 2x + 1, find the average rate of change from x = 1 to x = 3
Solution:
f(1) = 2(1) + 1 = 3, so Point 1: (1, 3)
f(3) = 2(3) + 1 = 7, so Point 2: (3, 7)
Rate = (7 - 3) / (3 - 1) = 4 / 2 = 2
For f(x) = x², find the average rate of change from x = 0 to x = 4
Solution:
f(0) = 0² = 0, so Point 1: (0, 0)
f(4) = 4² = 16, so Point 2: (4, 16)
Rate = (16 - 0) / (4 - 0) = 16 / 4 = 4
If a car travels 100 meters in 5 seconds, the average velocity is 100/5 = 20 m/s
If a company's revenue grows from $1M to $1.5M over 2 years, the average growth rate is $0.5M/2 years = $0.25M per year
If a bacteria population grows from 1000 to 8000 in 3 hours, the average growth rate is 7000/3 ≈ 2333 bacteria per hour
Find the average rate of change for f(x) = x² + 2x from x = -1 to x = 2
A ball is thrown upward. Its height h(t) = -16t² + 64t + 5 feet after t seconds. Find the average rate of change from t = 1 to t = 3 seconds.
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1. Choose between Coordinates or Function mode
2. Enter your data (points or function)
3. Click "Calculate Rate of Change"
4. View the detailed solution
• Use Function mode for mathematical expressions
• Check your signs carefully
• Remember: rate can be positive, negative, or zero
• Zero rate means no change on average