AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS
1.2 — Rates of Change
Notes — Average and Instantaneous
💡 Learning Objectives (1.2.A and 1.2.B) By the end of this lesson you will be able to:
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1. The Big Idea
How quickly does a quantity change as we vary another? That is the question rate of change addresses. In precalculus we study two flavours of rate:
- Average rate of change (AROC) over an interval — how much output changed per unit of input, on average, across that interval.
- Instantaneous rate of change (IROC) at a point — how quickly output is changing exactly at that input value.
AROC is what you can compute exactly with two function values. IROC is what AROC approaches as the interval shrinks down to a single point — the formal definition belongs to calculus, but in precalculus we approximate it numerically.
2. Average Rate of Change
💡 Definition For a function f and two distinct inputs a and b, the average rate of change of f over [a, b] is: AROC = ( f(b) − f(a) ) / ( b − a ) Geometrically, this is the slope of the secant line through the two points (a, f(a)) and (b, f(b)). |
The average rate of change over [1, 4] is the slope of the secant line connecting the two endpoints on the graph of f. Same numerical formula as the slope of any line.
📘 Example — Computing AROC For f(x) = 0.5x², find the average rate of change on [1, 4]. f(1) = 0.5, f(4) = 8. AROC = (8 − 0.5) / (4 − 1) = 7.5 / 3 = 2.5. Interpretation: on [1, 4], the function f rose at an average pace of 2.5 output units per input unit. |
2.1 Units
In contextual problems, the AROC always carries units of (output units) / (input units). For instance, if d is distance in metres and t is time in seconds, then AROC of d with respect to t has units of m/s — speed.
3. From AROC to IROC
To approximate the instantaneous rate of change at a point x = c, compute AROC on a small interval that contains c. The smaller the interval, the better the approximation.
As the interval shrinks around x = 2, the secant slopes get closer and closer to the slope of the tangent line at x = 2. That tangent slope is the instantaneous rate of change.
📘 Example — Estimating IROC numerically Estimate the IROC of f(x) = x² at x = 2 using progressively smaller intervals to the right. • AROC on [2, 3] = (9 − 4)/1 = 5. • AROC on [2, 2.5] = (6.25 − 4)/0.5 = 4.5. • AROC on [2, 2.1] = (4.41 − 4)/0.1 = 4.1. • AROC on [2, 2.01] = (4.0401 − 4)/0.01 = 4.01. The estimates approach 4 as the right endpoint slides toward 2. The true IROC at x = 2 is 4. |
⚠️ Centered intervals can be more accurate Estimating the rate at x = 2 using AROC on [1.99, 2.01] usually gives a sharper answer than using AROC on [2, 2.01], because the small interval is balanced around 2. |
4. Comparing Rates at Two Points
To decide whether f is changing faster at x = a or at x = b:
- Step 1: compute (or estimate) the IROC at x = a using a small interval around a.
- Step 2: do the same for x = b.
- Step 3: compare the absolute values — whichever is larger indicates faster change.
📘 Example — Where is the function changing faster? Where is f(x) = x² changing faster: at x = 1 or at x = 5? • AROC on [0.99, 1.01] = ( (1.01)² − (0.99)² )/0.02 = (1.0201 − 0.9801)/0.02 = 2. • AROC on [4.99, 5.01] = ( (5.01)² − (4.99)² )/0.02 ≈ 10. The function is changing roughly five times faster at x = 5 than at x = 1. |
5. The Sign of the Rate of Change
A positive rate of change means the two quantities grow together. A negative rate of change means one shrinks as the other grows. A zero rate of change means the output stays put for a moment.
💡 Reading the sign
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6. Putting It All Together
📘 Example — A real-world reading A snowboarder's distance (in meters) from the lodge t hours after sunrise is given by the table: t : 1, 2, 3, 4, 5, 6 d : 200, 350, 450, 510, 540, 555 • AROC on [1, 6] = (555 − 200)/(6 − 1) = 71 m/h. On average, she moved 71 m/h further from the lodge each hour. • AROC on [1, 2] = 150 m/h. AROC on [5, 6] = 15 m/h. Her speed (away from the lodge) is dropping over the day. • Estimate the rate at t = 4 using [3, 5]: (540 − 450)/2 = 45 m/h. |
7. Summary
- AROC = (Δoutput) / (Δinput) = the slope of the secant line through the two endpoints
- AROC always carries units = (output units) per (input units)
- To estimate IROC at x = c, use AROC over a small interval containing c
- Compare two IROCs to decide where a function is changing faster
- Sign of AROC tells you whether the function rose, fell, or returned to the same level