Which solutions can disappear when you divide?
You will be able to: Recover equilibrium solutions excluded during separation.
Which solutions can disappear when you divide?
A system with rate y′=y(2−y) stops changing at levels 0 and 2. Yet dividing by y(2−y) removes both values from the algebra. We must keep them in the solution set.
A useful starting point: How do you separate variables to find a solution family? →
Words and symbols before equations
- Equilibrium
- A constant solution with zero derivative.
- Excluded value
- A value disallowed by an algebraic division.
- Nonconstant branch
- A solution varying with the independent variable.
- Direct substitution
- Checking the original equation before transformations.
What this picture assumes
Original model; rounded readouts and finite sampled graphs. P′=0.5P(1−P/100), t in days. K=100 and r=0.5/day fixed; P is treated continuously. Teal dashed line is capacity. Exact curves include separate 0 and 100 equilibria. The rate graph distinguishes population from growth rate.
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.
- P₀=20; initial rate=8 organisms/day. At 6 days P=83.3925. Reaches 50 at t=2.77259 days; peak positive rate 12.5 organisms/day.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Before dividing, solve y(2−y)=0. The constant functions y=0 and y=2 both have derivative zero, matching the rate law.
For other values, dy/[y(2−y)]=dx. The identity 1/[y(2−y)]=(1/2)(1/y+1/(2−y)) allows integration.
Integrating carefully gives (1/2)ln∣y∣−(1/2)ln∣2−y∣=x+C. The minus sign in the second term comes from differentiating 2−y.
This logarithmic form applies only away from 0 and 2. It describes nonconstant branches on valid intervals; the two constant solutions must remain listed separately. The slope field explains their role even without solving explicitly.
A worked example, step by step
For y′=y(y−3), identify equilibria and the direction of motion between them.
- Set y(y−3)=0 to obtain y=0 and y=3.
- Check each constant in the original equation: both sides equal zero.
- For 0<y<3, y is positive and y−3 is negative, so y′<0.
- Thus these solutions decrease; division by y(y−3) would omit the equilibria and needs a separate check.
Canceling a factor can remove an entire constant solution. A logarithmic formula cannot be evaluated at a zero of its argument.
Does division proving a formula for y≠0 prove zero is impossible?
Compare with an explanation
No. It only restricts that algebraic derivation; test zero in the original equation.
Predict. Change one thing. Explain.
In the displayed logistic model, compare initial amounts 0, the carrying capacity and a value between. Explain which curves are constant and why the algebraic division would exclude them.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
P₀=20; initial rate=8 organisms/day. At 6 days P=83.3925. Reaches 50 at t=2.77259 days; peak positive rate 12.5 organisms/day.
Original model; rounded readouts and finite sampled graphs. P′=0.5P(1−P/100), t in days. K=100 and r=0.5/day fixed; P is treated continuously. Teal dashed line is capacity. Exact curves include separate 0 and 100 equilibria. The rate graph distinguishes population from growth rate.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the rate units, derivative signs, initial condition, domain or solution check. 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 questionAnalyze constant solutions and derivative signs for y′=(y−1)(4−y).
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Set each factor to zero: y=1 and y=4.
- 1 point: Verify both constant functions directly.
- 1 point: For 1<y<4 both factors are positive, so y increases.
- 1 point: Below 1 or above 4 the product is negative; these signs describe the field without requiring an explicit formula.
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 1When should equilibria be checked?
Before dividing by a function of y, and again against the original equation.
RECALL 2What is an equilibrium’s graph?
A horizontal solution curve.
RECALL 3Does separation always preserve all solutions?
No; division can exclude equilibria that must be restored.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Which solutions can disappear when you divide?
- For y′=g(y), check g(c)=0 for constant solutions y=c.
- Keep solutions excluded by division.
Remember: Canceling a factor can remove an entire constant solution. A logarithmic formula cannot be evaluated at a zero of its argument.
Conditions: Original model; rounded readouts and finite sampled graphs. P′=0.5P(1−P/100), t in days. K=100 and r=0.5/day fixed; P is treated continuously. Teal dashed line is capacity. Exact curves include separate 0 and 100 equilibria. The rate graph distinguishes population from growth rate.
Refresh Kid · AP Calculus BC Unit 7 · Objectives FUN-7.D · Review edition
Framework, scope and review status
Mapped to College Board CED Topic 7.6, FUN-7.D. CED and Fall 2026 clarifications checked September 17, 2026. All nine BC topics are included, with Euler’s method and logistic interpretation. The correction to FUN-7.B.2 is reflected: an equation may have infinitely many solutions. Focused explanations, examples, visual models and practice are original Refresh Kid work.
Rate and amount units, signs, initial data, lost equilibria, solution intervals and model assumptions are explicit. Slope fields and Euler polygons are finite illustrations; exact algebra supports identities and limits. The logistic explicit formula supports the visualization; required interpretation can be done from the rate law without deriving that formula. The smooth examples have unique solutions; no universal uniqueness claim is made.
Organic Chemistry Tutor video titles and destinations were checked, not the full videos. Khan Academy’s destination was checked; its JavaScript lesson contents were not fully readable by the research tool. OpenStax sections 4.1–4.4 supplied conceptual cross-checks. No provider questions, artwork or scripts were copied. No affiliation or endorsement is implied.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. The original tank reuses self-hosted Three.js with its MIT license retained. The 2×2 dm base connects water depth to volume: 1 dm³=1 L. This tank uses a feedback-controlled pump, not a gravity-drain law. Its 3D height and labeled 2D graphs use the same exact exponential formula. Rotation changes only the view, and complete explanations remain available without WebGL.
Independent teacher review and student usability testing remain pending. Technical checks do not certify mathematical accuracy, full accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. This multi-unit archive is not assigned as a complete Unit 7 task.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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