Mathematics
Grade 12
15 min
Average rate of change II
Average rate of change II
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1
Introduction & Learning Objectives
Learning Objectives
Calculate the average rate of change of a non-linear function over a specified interval.
Formulate and simplify the difference quotient for polynomial, rational, and radical functions.
Interpret the average rate of change as the slope of the secant line connecting two points on a curve.
Relate the average rate of change to real-world scenarios, such as average velocity.
Analyze the effect of shrinking the interval on the value of the average rate of change.
Distinguish between the average rate of change and the value of the function at a point.
A sports car accelerates from 0 to 60 mph in 3 seconds. What was its *average* acceleration during that time? 🏎️💨
This lesson builds on your understanding of slope to explore how non-linear functions change on av...
2
Key Concepts & Vocabulary
TermDefinitionExample
Average Rate of Change (AROC)A measure of how much a function's output (y-value) changes, on average, for each unit of change in its input (x-value) over a specific interval.If a car travels 150 km in 2 hours, its average rate of change (average speed) is 150 km / 2 hours = 75 km/h.
Secant LineA straight line that intersects a curve at two distinct points. The slope of the secant line is equal to the average rate of change of the function between those two points.For the function f(x) = x^2, the secant line connecting the points (1, 1) and (3, 9) has a slope of (9-1)/(3-1) = 4.
Interval [a, b]The set of all real numbers between and including 'a' and 'b'. For AROC, this represents the domain over which the change is being measured.The interval...
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Core Formulas
Average Rate of Change Formula (Interval Form)
AROC = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}
Use this formula when you are given a specific interval [a, b]. It calculates the slope of the secant line between the points (a, f(a)) and (b, f(b)).
Average Rate of Change Formula (Difference Quotient Form)
AROC = \frac{f(x+h) - f(x)}{h}
This is a more general form where the interval is from x to x+h, and the length of the interval is h. This form is crucial for the definition of the derivative.
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Easy
What does the average rate of change of a function over an interval [a, b] represent geometrically?
A.The slope of the tangent line at x=a
B.The slope of the secant line connecting (a, f(a)) and (b, f(b))
C.The area under the curve between a and b
D.The y-intercept of the function
Easy
Which formula is used to calculate the average rate of change of a function f(x) over a specific interval [a, b]?
A.\frac{f(x+h) - f(x)}{h}
B.f(b) - f(a)
C.\frac{f(b) - f(a)}{b - a}
D.b - a
Easy
What is the average rate of change of the linear function f(x) = 7x - 3 over the interval [2, 5]?
A.21
B.7
C.3
D.14
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