How does an interval become a rate?
You will be able to: Build a signed difference quotient and interpret its units.
How does an interval become a rate?
A toy vehicle is s(t)=t²+1 meters from a marker. At 2 seconds it is 5 meters away; at 3 seconds it is 10 meters away. Its change in position is 5 meters during 1 second.
A useful starting point: Prerequisite: average change, inputs, outputs and slope →
Words and symbols before equations
- Δ
- Final minus initial, read as change in.
- Difference quotient
- Output change divided by nonzero input change.
- Secant line
- Line through two distinct graph points.
- h
- Signed change in input; here measured in seconds.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. s(t)=t²+1 meters; target t=2 seconds. h=±10⁻ᵖ is never zero. The position graph is not a physical path.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- h=0.1 s; rise=0.41 m; run=0.1 s; secant rate=4.1 m/s. Exact derivative at 2 s=4 m/s. Blue: position; orange: secant. Position s(2)=5 m is a different quantity.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Use the same order in numerator and denominator: [s(b)−s(a)]/(b−a). Reversing both orders leaves the rate unchanged; reversing only one changes its sign incorrectly.
Writing b=a+h gives [s(a+h)−s(a)]/h. At a=2 this is [(2+h)²+1−5]/h=(4h+h²)/h=4+h for h≠0.
The graph’s vertical axis is position, not speed. Its secant slope has units meters per second because output units divide by input units.
A negative h samples an earlier point. Both rise and run can be negative even when the quotient is positive. An interval average is not automatically the instantaneous rate.
A worked example, step by step
Find the average rate from t=2 to t=1.5 seconds.
- Identify h=1.5−2=−0.5 s.
- Compute s(1.5)=3.25 m and s(2)=5 m.
- Divide (3.25−5)/(1.5−2)=−1.75/−0.5=3.5 m/s.
- The positive quotient agrees with motion toward increasing position as time increases.
Do not mix the order of subtraction or confuse position height with slope.
Does negative h force a negative average rate?
Compare with an explanation
No. For this increasing position rule both changes are negative on a left interval, so their quotient is positive.
Predict. Change one thing. Explain.
Choose left and right approaches. Before moving the zoom, predict the signs of rise and run and whether their quotient is positive.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
h=0.1 s; rise=0.41 m; run=0.1 s; secant rate=4.1 m/s. Exact derivative at 2 s=4 m/s. Blue: position; orange: secant. Position s(2)=5 m is a different quantity.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. s(t)=t²+1 meters; target t=2 seconds. h=±10⁻ᵖ is never zero. The position graph is not a physical path.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. 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 questionFind and interpret the average rate of s(t)=t²+1 from 2 to 2.5 s. Repeat with both differences reversed.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: s(2)=5 m and s(2.5)=7.25 m.
- 1 point: Position change=2.25 m; time change=0.5 s.
- 1 point: Average rate=4.5 m/s.
- 1 point: Reversed differences give −2.25/−0.5=4.5 m/s, the same 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 is a secant slope?
Average output change per input change between two points.
RECALL 2Why must h be nonzero?
The quotient divides by h.
RECALL 3What units does ds/dt have?
Position units divided by time units, here m/s.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does an interval become a rate?
- Average rate=[f(b)−f(a)]/(b−a), b≠a.
- Equivalent increment form: [f(a+h)−f(a)]/h, h≠0.
Remember: Do not mix the order of subtraction or confuse position height with slope.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. s(t)=t²+1 meters; target t=2 seconds. h=±10⁻ᵖ is never zero. The position graph is not a physical path.
Refresh Kid · AP Calculus AB Unit 2 · Objectives CHA-2.A, CHA-2.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.1, CHA-2.A, CHA-2.B. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were 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. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
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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