2.6 Average Rate of Change and Limits Intro
Key Takeaways
- Average rate of change on [a, b] equals [f(b) - f(a)] / [b - a], the slope of the secant line.
- For f(x) = x² - 1 on [1, 4], f(1) = 0 and f(4) = 15, so the average rate of change is 5.
- A limit describes the value f(x) approaches as x nears a point, which may differ from the function value at a hole.
- If left-hand and right-hand limits differ at a point, the two-sided limit does not exist.
- Praxis 5165 links average rates to derivative intuition and instructional activities with shrinking secant intervals.
Why This Section Matters
Average rate of change and introductory limits bridge functions to calculus on Praxis 5165. ETS tests whether you compute secant slopes, interpret approached values from tables and graphs, and explain continuity — often in task-of-teaching items linking rates to slope and motion contexts.
Average Rate of Change
On [a, b], the average rate of change of f is:
[f(b) - f(a)] / [b - a]
This equals the slope of the secant line through (a, f(a)) and (b, f(b)).
Worked Example: Quadratic on [1, 4]
For f(x) = x² - 1:
f(1) = 0, f(4) = 15
Average rate = (15 - 0)/(4 - 1) = 5
Interpretation: over that interval, output increases an average of 5 units per unit increase in input. This is not the instantaneous rate at any single point.
Linear Check
For f(x) = 2x + 3 on [0, 4]: (11 - 3)/4 = 2, matching slope — when rate of change is constant, average rate equals instantaneous rate everywhere.
Toward Instantaneous Rate: Limits
The derivative at a is the limit of average rates as the interval shrinks:
f'(a) = lim_{h→0} [f(a + h) - f(a)]/h
Praxis 5165 does not require formal epsilon-delta proofs, but you must understand a limit as the value f(x) approaches as x nears a point — possibly different from the function value at that point.
Estimating Limits
| Graph Feature | Limit Behavior |
|---|---|
| Removable hole | Limit may exist though f(a) is undefined or differs |
| Jump | Left and right limits differ → two-sided limit DNE |
| Vertical asymptote | Function grows without bound |
| Continuous at a | lim_{x→a} f(x) = f(a) |
Worked Example: Hole vs. Value
A table shows f(x) → 3 as x → 2 from both sides, but f(2) is undefined. The limit as x → 2 is 3 (removable discontinuity). The graph has a hole at (2, 3).
Worked Example: Piecewise Jump
f(x) = { x + 1, x < 2; 5, x = 2; x², x > 2 }
- Left limit at 2: 3
- Right limit at 2: 4
- Two-sided limit does not exist at x = 2
Teaching: Secants Approaching a Tangent
A strong instructional activity: compute secant slopes on f(x) = x² over [1, 1 + h] for h = 1, 0.5, 0.1 near x = 1. Slopes approach 2, estimating the tangent slope (derivative) at x = 1. This connects average rate to instantaneous rate before formal differentiation rules.
Motion Context
If s(t) is position, average velocity on [t₁, t₂] is [s(t₂) - s(t₁)]/(t₂ - t₁). Instantaneous velocity at time a is s'(a). Displacement vs. total distance distinctions appear occasionally — distance accounts for direction changes.
Notation and Praxis Logistics
Limit notation lim_{x→a} f(x) = L means values of f(x) get arbitrarily close to L as x approaches a (without requiring f(a) = L). One-sided limits use x → a⁻ or x → a⁺.
Common Praxis Traps
- Reporting f(b) instead of average rate
- Confusing limit with function value at a hole
- Assuming a limit exists at a jump
- Using (b - a) incorrectly in the denominator
Section Takeaways
Compute average rate with the difference quotient, interpret limits from graphs and tables, and connect shrinking secants to derivative intuition. Practice on /practice/praxis-math rate-of-change items.
Difference Quotient Notation
The expression [f(x + h) - f(x)]/h is the difference quotient. Average rate on [a, b] is the difference quotient with x = a and h = b - a.
For f(x) = x², on [1, 1 + h]:
[f(1 + h) - f(1)]/h = [(1 + 2h + h²) - 1]/h = (2h + h²)/h = 2 + h
As h → 0, the value approaches 2 — the instantaneous rate at x = 1.
Continuity at a Point
f is continuous at x = a when all three hold:
- f(a) is defined
- lim_{x→a} f(x) exists
- lim_{x→a} f(x) = f(a)
Polynomials are continuous everywhere. Rational functions are continuous on their domains.
Infinite Limits and Asymptotes
As x → 3⁺ for f(x) = 1/(x - 3), outputs grow toward +∞ — an infinite limit, not a finite limit value. This behavior supports vertical asymptote identification, linking back to rational functions in Section 2.3.
Average vs. Instantaneous in Motion
If s(t) = t² meters, average velocity on [1, 3] is (9 - 1)/(3 - 1) = 4 m/s. Instantaneous velocity at t = 2 is s'(2) = 4 m/s — equal here by coincidence on a quadratic, not a general rule.
Limit Laws (Informal)
If lim f(x) = L and lim g(x) = M, then limits of sums, differences, products, and quotients (when M ≠ 0) combine predictably. Praxis items may ask whether a limit can be found by direct substitution — possible when the function is continuous at that point.
One-Sided Limits
lim_{x→2⁻} f(x) approaches from the left; lim_{x→2⁺} f(x) from the right. A two-sided limit exists only when both one-sided limits exist and are equal.
Piecewise definitions at boundaries are the standard Praxis setup for testing this idea.
Relating Limits to ETS Calculator Use
Tables generated on the on-screen calculator can estimate limits numerically. If f(1.9) ≈ 3.61, f(1.99) ≈ 3.9601, and f(2.01) ≈ 4.0401, outputs approach 4 as x → 2 even if f(2) is undefined — reinforcing removable discontinuity reasoning.
What is the average rate of change of f(x) = x² - 1 on the interval [1, 4]?
A teacher wants students to connect derivatives to slope. Which activity best supports that goal?