LEARN · EXPLAIN · REVISE

Compare the changes over equal unit intervals for L(t)=3t and E(t)=3·2ᵗ.

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

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

Hint 2

Compare differences for one model and quotients for the other.

Worked solution

Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$L(t+1)-L(t)=3;\quad E(t+1)/E(t)=2$
Equal ratios and equal differences describe different mechanisms.

L has constant increments; E has constant ratios.

Checks and common pitfalls: Equal ratios and equal differences describe different mechanisms.

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.