LEARN · EXPLAIN · REVISE

A log-transformed data plot looks straight. Does this prove the original process is exponential at all times? Explain.

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 / Transfer#Worked example

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the context into an exponential or logarithmic equation and check what the model assumes.

Hint 2

Separate descriptive fit from causal evidence.

Worked solution

Translate the context into an exponential or logarithmic equation and check what the model assumes.
Inspect residuals, measurement error, alternative models and the time interval before using predictions.
Positive data are also required for a real logarithmic transform.

No; it supports a model within the observed range but does not prove the mechanism or unlimited extrapolation.

Checks and common pitfalls: Positive data are also required for a real logarithmic transform.

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.