LEARN · EXPLAIN · REVISE

A continuous population model predicts 121.5 individuals. Explain how to report this result responsibly.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 4.5 · PDF 149 / printed page 142

TOPIC 01

Applications of functions II

Build understanding of applications of functions ii through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish the model output from the measured quantity.

Hint 2

Think about what the unit “individual” permits.

Worked solution

Distinguish the model output from the measured quantity.
The model is continuous but the actual count is discrete; a decimal prediction is not an observation of a fractional individual.
Rounding does not remove model error or establish exactness.

Treat 121.5 as a model estimate, state a rounding rule if a whole count is required, and retain uncertainty.

Checks and common pitfalls: Rounding does not remove model error or establish exactness.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Interpret and use applications of functions ii with domain checks.
Multiplicative change: Constant ratios lead to exponential models; logarithms solve elapsed-time questions.
Zeros and evidence: Continuity plus a sign change gives a zero; model validity still needs evidence.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Constant ratios lead to exponential models; logarithms solve elapsed-time questions.
Expected reasoning: Continuity plus a sign change gives a zero; model validity still needs evidence.
Expected correction: A fitted trend does not justify unlimited extrapolation.