Mathematics
Grade 12
15 min
Average rate of change I
Average rate of change I
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1
Introduction & Learning Objectives
Learning Objectives
Define the average rate of change of a function over an interval.
Calculate the average rate of change from a function given its equation.
Determine the average rate of change from a table of values.
Find the average rate of change from a graph.
Explain the geometric interpretation of the average rate of change as the slope of a secant line.
Interpret the meaning of the average rate of change in the context of real-world applications.
Ever wonder what your average speed was on a road trip, even if you sped up and slowed down? 🚗 That's an average rate of change!
This lesson introduces the foundational concept of the average rate of change, which measures how a function's output changes, on average, relative to its input over a specific interval...
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Key Concepts & Vocabulary
TermDefinitionExample
Rate of ChangeA measure of how one quantity changes with respect to the change in another quantity. It describes the 'steepness' of the relationship between two variables.Speed is a rate of change, describing the change in distance with respect to the change in time.
IntervalA range of values for a variable, typically the independent variable (x). For average rate of change, we consider a closed interval, denoted as [a, b].The time interval from 2 seconds to 5 seconds is written as [2, 5].
Average Rate of Change (AROC)The ratio of the total change in the function's output (dependent variable) to the change in its input (independent variable) over a given interval.If a car travels 150 km in 2 hours, its average rate of change (average speed) is 150 km /...
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Core Formulas
Average Rate of Change Formula
AROC = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}
Use this formula to calculate the average rate of change of a function f(x) over the interval [a, b]. First, evaluate the function at the endpoints of the interval, f(a) and f(b). Then, find the difference in the function values and divide by the difference in the x-values.
Slope of a Secant Line
m_{sec} = \frac{y_2 - y_1}{x_2 - x_1}
This is the geometric equivalent of the average rate of change formula. Given two points on a curve, (x1, y1) and (x2, y2), this formula calculates the slope of the line connecting them. This slope is exactly the average rate of change between x1 and x2.
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Challenging
The average rate of change of the function f(x) = x^3 on the interval [1, b] is 7. Find the value of b.
A.3
B.2
C.√7
D.6
Challenging
For a function f(x) that is strictly increasing on the interval (a, b), what must be true about its average rate of change on any sub-interval [x1, x2] where a < x1 < x2 < b?
A.The average rate of change must be positive.
B.The average rate of change must be negative.
C.The average rate of change must be zero.
D.The average rate of change could be positive, negative, or zero.
Challenging
If the average rate of change of a function f(x) on the interval [a, b] is zero, what can be definitively concluded?
A.The function f(x) must be a constant function on [a, b].
B.The graph of the function has a horizontal tangent line between a and b.
C.f(a) = f(b).
D.The function must be f(x) = 0.
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