Defining Average and Instantaneous Rates of Change at a Point
Build a finite instantaneous rate from either form of the limiting secant quotient.
College Board Topic 2.1 · CED p. 55 (PDF p. 60) · CED effective Fall 2020; project course outline 2026–27.
Two points give a secant slope, the average rate [f(b)−f(a)]/(b−a). One point can have an instantaneous rate when nearby secant slopes approach a finite common value. Write that value as f′(a). Its units are output units per input unit.
The derivative at a can be written in two equivalent ways: f′(a)=lim(h→0)[f(a+h)−f(a)]/h or f′(a)=lim(x→a)[f(x)−f(a)]/(x−a). Setting x=a+h connects them. The limit must exist as a finite real number for this derivative to exist.
For s(t)=t² meters, the average speed from 1 to 3 seconds is (9−1)/2=4 m/s. At t=1, the difference quotient is 2+h, so the instantaneous speed is 2 m/s. The average over a long interval does not describe every instant inside it.
Let's work through it
- For f(x)=x² at a=3, form [(3+h)²−9]/h. Reason: the derivative is a limit of secant slopes.
- For h≠0, simplify to 6+h. Reason: expansion exposes the common factor h.
- Let h→0 to find f′(3)=6. Reason: both sides approach the same finite slope.
Try it from memory
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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
What is the average rate of x² from 2 to 5?
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Retrieval 1
What is the average rate of x² from 2 to 5?
Worked answer and reason
7.
The secant slope is (25−4)/(5−2).
Retrieval 2
Which substitution connects the h and x derivative definitions?
Worked answer and reason
x=a+h.
Then x−a=h and approaching x=a is equivalent to approaching h=0.
Catch the tempting mistake
- The average and instantaneous rates at an interval endpoint are always equal.
- They describe different quantities; equality happens for special models such as a constant slope.
Make the picture move
Explore → predict → explain
A derivative is a limit of slopes.
Try first. At the selected a, predict the tangent slope. Which side’s secant lies above it, and which lies below?
Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.
- Function: x²
- Selected secant
- Exact tangent
Right secant slope 2.5 at a = 1. The derivative is 2.
Compare the numbers
| Distance |h| | Left secant slope | Right secant slope |
|---|---|---|
| 0.5 | 1.5 | 2.5 |
| 0.25 | 1.75 | 2.25 |
| 0.1 | 1.9 | 2.1 |
| 0.05 | 1.95 | 2.05 |
What’s going on? For x², the difference quotient simplifies to 2a + h. As h approaches zero from either side, the secant slope approaches 2a.
Keep in mind. The controls keep h nonzero so that the quotient is defined. A small h is an approximation; the limiting calculation supplies the exact derivative.
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For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
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- Unit 2 problem 1 · No calculator · verbal
- Unit 2 problem 2 · No calculator · analytic
- Unit 2 problem 3 · No calculator · verbal
Explain and apply the mathematics
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Represent the interval rate and its limiting tangent
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For the teacher
- Keep secant and tangent vocabulary attached to their formulas.
- Ask for units and the interpretation before arithmetic.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Instantaneous rate of change
An instantaneous rate of change is the limiting average rate around one input when that limit exists. For a differentiable function, it is its derivative at that input. For example, position s(t)=t² meters has instantaneous rate 4 meters per second at t=2 seconds. The rate’s units are output units per input unit.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Difference quotient
A difference quotient divides a change in a function’s output by the associated nonzero input change. The form [f(a+h)−f(a)]/h is an average rate over an interval of length h. For example, for f(x)=x² it simplifies to 2a+h when h≠0. Taking its limit as h approaches 0 defines the derivative at a if the finite limit exists.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Power rule
The derivative power rule gives d(xⁿ)/dx=n xⁿ⁻¹ wherever the real power function is differentiable. Constant multiples remain as constant factors. For example, the derivative of 3x⁴ is 12x³. A composite power such as (x²+1)³ also requires the chain rule.