What constant value would give the same total?
You will be able to: Calculate average function value and distinguish it from average rate of change.
What constant value would give the same total?
A room warms during the morning. Its average temperature should account for all the time spent at each temperature, rather than just the first and last readings.
A useful starting point: Prerequisite: definite integrals and endpoint evaluation →
Words and symbols before equations
- Average value
- Constant height with the same signed integral over the interval.
- Interval width
- b−a, with a<b.
- Average rate of change
- Endpoint change divided by interval width.
- Integral
- Signed accumulation of function value times input width.
What this picture assumes
Original model; numerical labels are rounded. T(t)=20+t² °C on [0,b]. Average is 20+b²/3 °C. The dashed average line defines a rectangle with the same area; average rate instead has units °C/hour.
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.
- b=2 h; average temperature 21.3333 °C; accumulated temperature 42.6667 °C·h; average rate 2 °C/h. Dashed teal line is the average; orange line is the upper bound.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
If the constant height is M, its integral on [a,b] is M(b−a). Setting this equal to the integral of f gives M=(1/(b−a))∫ₐᵇ f(x)dx.
For T(t)=20+t² °C on [0,2] hours, the integral is 40+8/3 °C·h. Dividing by 2 h gives 64/3≈21.333 °C.
The endpoint temperature average is 22 °C, different because the curve is not linear. The average rate of temperature change is (24−20)/2=2 °C/h, a different quantity with different units.
For continuous f, at least one input attains its average value. Solve f(c)=M if asked for such an input; do not confuse this with the derivative-based Mean Value Theorem.
| Feature | Average value | Average rate |
|---|---|---|
| Operation | Integral divided by width | Endpoint change divided by width |
| Temperature units | Degrees | Degrees per hour |
| What it describes | Typical function height | Net change per input unit |
A worked example, step by step
Find the average value of f=x² on [1,3] and an input where it is attained.
- The interval width is 2.
- The integral is [x³/3]₁³=26/3.
- Divide by 2 to get average value 13/3.
- Solve c²=13/3; c=√(13/3) is in [1,3], so it attains the average.
Average value uses an integral; average rate uses endpoint change. An endpoint average is not generally the average function value.
If a temperature integral has units °C·h, what units does its average have?
Compare with an explanation
Divide by interval length in hours to obtain °C.
Predict. Change one thing. Explain.
Move the upper time b from 1 to 4. Compare the shaded temperature accumulation with the equal-area constant rectangle. Explain why its height is neither endpoint temperature.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
b=2 h; average temperature 21.3333 °C; accumulated temperature 42.6667 °C·h; average rate 2 °C/h. Dashed teal line is the average; orange line is the upper bound.
Original model; numerical labels are rounded. T(t)=20+t² °C on [0,b]. Average is 20+b²/3 °C. The dashed average line defines a rectangle with the same area; average rate instead has units °C/hour.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the units, signed change, strip direction, radius distances or cross-sectional area. 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 the average of 3x²+1 on [0,2], its average rate, and a point attaining the average value.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The integral is [x³+x]₀²=10.
- 1 point: Average value is 10/2=5.
- 1 point: Average rate is (13−1)/2=6.
- 1 point: 3c²+1=5 gives c=2/√3 in [0,2].
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 1Why divide by b−a?
To turn accumulated value into an equal-area rectangle height.
RECALL 2What are average-value units?
The original function units.
RECALL 3Does averaging endpoints always work?
No; it works for a linear function but not in general.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What constant value would give the same total?
- f_avg=(1/(b−a))∫ₐᵇ f(x)dx, a<b.
- Average value has the units of f.
- For continuous f, solve f(c)=f_avg to locate an attaining input.
Remember: Average value uses an integral; average rate uses endpoint change. An endpoint average is not generally the average function value.
Conditions: Original model; numerical labels are rounded. T(t)=20+t² °C on [0,b]. Average is 20+b²/3 °C. The dashed average line defines a rectangle with the same area; average rate instead has units °C/hour.
Refresh Kid · AP Calculus AB Unit 8 · Objectives CHA-4.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.1, CHA-4.B. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. Unit 8 covers AB topics 8.1–8.12. Topic 8.13 (arc length) is BC-only. Focused lesson titles, examples, questions and illustrations are original Refresh Kid material. Unit 7 is a separate curriculum outline, not a prerequisite gate to these lessons.
Average value is distinguished from average rate. Motion uses velocity for displacement and speed for distance, with initial values stated separately. Area bounds and ordering are checked, including multiple crossings. Volumes derive the slice area before integration, distinguish diameter from radius, and use distances from the specified axis. Disks and washers use perpendicular slices and a consistent integration variable. A region crossing an axis requires checking the actual swept disk rather than inventing a hole.
The Organic Chemistry Tutor Disk & Washer Method video title, creator and relevant description were checked; the full video was not reviewed. Use the free video as an optional companion; no paid material is required. Khan Academy’s unit destination was checked, but its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 6.1 and 6.2 were consulted for conceptual cross-checking. No provider scripts, questions, artwork or diagrams were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. All cross-section and revolution geometry is original and uses existing self-hosted Three.js with its MIT license. Spatial coordinates preserve the mathematical lengths; sampled mesh surfaces illustrate exact formulas. Teal shows the solid and orange a selected zero-thickness section. The volume is for the entire solid, not the highlighted plane. No autoplay is used; camera rotation changes only the view. Labeled 2D regions, cross-section diagrams, readouts and calculations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical 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 here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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