Introducing Calculus: Can Change Occur at an Instant?
Explain instantaneous change as a limit of average rates, keeping amount and rate units separate.
College Board Topic 1.1 · CED p. 33 (PDF p. 38) · CED effective Fall 2020; project course outline 2026–27.
A speedometer seems to know your speed right now. A two-point calculation only knows your average speed over a stretch of time. Calculus bridges that gap: shrink the time interval around the instant and ask whether those average rates approach one number.
For position s(t), the average rate from t=a to t=a+h is [s(a+h)−s(a)]/h. Its units are position units per time unit. h can be negative: that interval samples the motion just before a. At h=0 the quotient would divide by zero; the instantaneous rate comes from a limit as h approaches 0, rather than substituting h=0.
Suppose s(t)=t² meters for t seconds. Near t=2, the average rate is [(2+h)²−4]/h=4+h for h≠0. Those rates approach 4 meters per second from both sides. The position is 4 meters; the rate is 4 meters per second. Equal numbers do not make them the same quantity.
Let's work through it
- Compute the average rate from t=2 to t=2.1: (4.41−4)/0.1=4.1 m/s. Reason: average rate is change in position divided by elapsed time.
- Use a variable interval: [(2+h)²−4]/h=4+h for h≠0. Reason: expansion and cancellation produce an equivalent quotient on the punctured interval.
- Let h approach 0: 4+h approaches 4. Reason: a limiting rate describes the instant without dividing by zero.
Try it from memory
Pause & retrieve
Try it before you peek.
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
If s(t)=3t², what is the average rate from 1 to 1+h?
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These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
If s(t)=3t², what is the average rate from 1 to 1+h?
Worked answer and reason
6+3h for h≠0.
Expand the difference before dividing by h.
Retrieval 2
Why does setting h=0 in the original quotient fail?
Worked answer and reason
The quotient becomes 0/0; the instantaneous rate requires a limit.
An interval of zero length has no average-rate quotient.
Catch the tempting mistake
- Instantaneous rate is the change in position at one point.
- A rate compares change with elapsed time; a limiting process is needed at an instant.
Make the picture move
Explore → predict → explain
Getting close is different from landing there.
Try first. Predict the limit at x = 1. Does changing the value at one point change what nearby values approach?
Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.
- Nearby values: y = x + 1
Both sides approach 2, although the filled point gives f(1) = 3. The limit exists; continuity fails.
Compare the numbers
| Distance h | x from the left | f(1 − h) | f(1 + h) |
|---|---|---|---|
| 0.5 | 0.5 | 1.5 | 2.5 |
| 0.25 | 0.75 | 1.75 | 2.25 |
| 0.1 | 0.9 | 1.9 | 2.1 |
| 0.05 | 0.95 | 1.95 | 2.05 |
What’s going on? The open circle is a missing point on that branch; the filled circle is the actual function value. A limit concerns all sufficiently nearby inputs, not just the value at the input itself.
Keep in mind. A finite table is evidence for a conjecture, not a proof of a limit. Here the branch formulas justify the exact one-sided limits.
Calculator workspace and limits
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.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 1 problem 1 · No calculator · verbal
For the teacher
- Have students describe the units before calculating.
- Use intervals on both sides of an instant; a one-sided trend alone cannot establish a two-sided limiting rate.
Glossary and deeper reading · 3 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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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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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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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.