How can we describe change at one instant?
You will be able to: Estimate an instantaneous rate using average rates on shrinking intervals.
How can we describe change at one instant?
A toy cart has position s(t)=t² meters after t seconds. From 2 to 3 seconds it moves 5 meters, but that whole-second average does not tell us its exact rate at 2 seconds.
A useful starting point: Unit 1 outline; review function inputs, outputs and slope below →
Words and symbols before equations
- Function
- A rule assigning one output to each allowed input.
- Δ, change
- Final value minus initial value.
- Average rate
- Change in output divided by the nonzero change in input.
- Secant
- A line through two distinct points on a graph.
- h
- A nonzero signed time interval in seconds.
What this picture assumes
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. s(t)=t² m for 0≤t≤4 s. h is never zero. The graph is position versus time, not a physical path. The optional 3D view keeps all coordinates in one plane with independent axis scales.
Read the picture in three steps
- Read the species and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- h=1 s; P=(2 s, 4 m), Q=(3 s, 9 m). Rise=5 m; run=1 s; average rate=5 m/s. The limit is 4 m/s as h→0; h is never zero. Blue: s=t². Orange: secant. Dashed: signed changes.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the chemistry
On a position–time graph, horizontal distance is elapsed time and vertical height is position. The graph is not the cart’s physical path. A rise/run triangle encodes displacement divided by time.
At t=2 and t=2+h, the average rate is [(2+h)²−4]/h. Expanding gives (4h+h²)/h=4+h for h≠0. The units are meters per second.
As h approaches zero from either side, 4+h approaches 4. The limiting average rate is 4 m/s. We never divide by a zero time interval; we study what nonzero intervals approach.
The optional rotatable graph keeps all points in one plane. Inspect the horizontal run and vertical rise, then use the numeric axis scales: apparent screen angle is not a reliable slope measurement.
A worked example, step by step
Estimate the rate at t=2 using h=0.5 s and h=0.1 s, then identify the limit.
- For h=0.5, the positions are 4 and 6.25 m.
- Average rate=(6.25−4)/0.5=4.5 m/s.
- For h=0.1, average rate=(4.41−4)/0.1=4.1 m/s.
- Since the simplified rate is 4+h, its limit as h→0 is 4 m/s.
A slope is a ratio of changes, not the height of the graph. Do not substitute h=0 into the original quotient.
Is s(2)=4 m the same quantity as the limiting rate 4 m/s?
Compare with an explanation
No. Their numbers happen to match, but position and rate have different meanings and units.
Predict. Change one thing. Explain.
Shrink the interval from the right, then from the left. Compare the rise/run quotient with the position height. Rotate the optional graph to inspect its plane and reset the view.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
h=1 s; P=(2 s, 4 m), Q=(3 s, 9 m). Rise=5 m; run=1 s; average rate=5 m/s. The limit is 4 m/s as h→0; h is never zero. Blue: s=t². Orange: secant. Dashed: signed changes.
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. s(t)=t² m for 0≤t≤4 s. h is never zero. The graph is position versus time, not a physical path. The optional 3D view keeps all coordinates in one plane with independent axis scales.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using nearby function values, graph coordinates, or the hypotheses of a limit law or theorem. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor s(t)=t², expand [s(3+h)−s(3)]/h for h≠0. State its limit and units, then explain why this does not divide by zero.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Expand to (6h+h²)/h.
- 1 point: Simplify to 6+h for h≠0.
- 1 point: The limit is 6 m/s.
- 1 point: Only nonzero intervals are evaluated; the limiting value defines the instantaneous rate.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does Δ mean?
Final minus initial.
RECALL 2What are slope units on a position–time graph?
Meters per second.
RECALL 3Does a rotated graph change the rate?
No. Coordinates and their ratio remain the same.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can we describe change at one instant?
- Average rate=[s(2+h)−s(2)]/h, h≠0.
- For s(t)=t², the rate near t=2 is 4+h; its limit is 4 m/s.
Remember: A slope is a ratio of changes, not the height of the graph. Do not substitute h=0 into the original quotient.
Conditions: Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. s(t)=t² m for 0≤t≤4 s. h is never zero. The graph is position versus time, not a physical path. The optional 3D view keeps all coordinates in one plane with independent axis scales.
Refresh Kid · AP Calculus AB Unit 1 · Objectives CHA-1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.1, CHA-1.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 1 has 16 official topics. Focused lesson titles and questions are original Refresh Kid teaching material. Topics 1.7 and 1.9 integrate earlier objectives and skills rather than introducing new numbered learning objectives.
Formal epsilon-delta proofs are not assessed in this unit. L’Hôpital’s rule, derivative rules and differentiation tests belong later in the course and are not used to bypass limit reasoning here. Trigonometric limits use radians. Finite graphs and tables provide evidence, not a proof of all nearby behavior. Infinity describes unbounded behavior, not a number to substitute. The Intermediate Value Theorem requires continuity on a closed interval and an intermediate output; it guarantees existence, not uniqueness.
The Organic Chemistry Tutor video title, creator and description were checked; the full video was not reviewed. Khan Academy’s current course announcement and linked unit destination were checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Section 2.2 was consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. Camera rotation does not add a variable or change slope; use labeled coordinates and axis scales.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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