Kenneth ChengMathematical modeling
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Applications of functions II

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

Revisit first: Logarithmic functions

TOPIC 01

Applications of functions II

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

What you will be able to explain

  • Interpret and use applications of functions ii with domain checks.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in applications of functions ii changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Exponential graph: horizontal shift 0, vertical shift 0. Input is replaced by x−0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

A population is 200 and increases by 10% each year. Find it after two years.

  • Use the domain, units and sampling assumptions stated in the question.
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
The second percentage applies to the enlarged population.
Worked solution

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

  2. Calculate or simplify this relation.

    200(1.1)2=242200(1.1)^2=242

  3. This is compounding, not a single 20% increase.

The requested value is 242.

Checks and common pitfalls: This is compounding, not a single 20% increase.

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

02 / Standard#Worked example

A sample starts at 160 mg and halves every 3 days. Find its mass after 9 days.

  • Use the domain, units and sampling assumptions stated in the question.
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
Nine days contains three half-lives.
Worked solution

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

  2. Calculate or simplify this relation.

    160(1/2)9/3=20160(1/2)^{9/3}=20

  3. Use matching time units in the exponent.

The requested value is 20.

Checks and common pitfalls: Use matching time units in the exponent.

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

03 / Transfer#Worked example

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

  • Use the domain, units and sampling assumptions stated in the question.
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

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

  2. Inspect residuals, measurement error, alternative models and the time interval before using predictions.

  3. 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
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

A population is 300 and increases by 10% each year. Find it after two years.

  • Use the domain, units and sampling assumptions stated in the question.
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
The second percentage applies to the enlarged population.
Worked solution

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

  2. Calculate or simplify this relation.

    300(1.1)2=363300(1.1)^2=363

  3. This is compounding, not a single 20% increase.

The requested value is 363.

Checks and common pitfalls: This is compounding, not a single 20% increase.

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

05 / Foundation#Your turn

A sample starts at 240 mg and halves every 3 days. Find its mass after 9 days.

  • Use the domain, units and sampling assumptions stated in the question.
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
Nine days contains three half-lives.
Worked solution

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

  2. Calculate or simplify this relation.

    240(1/2)9/3=30240(1/2)^{9/3}=30

  3. Use matching time units in the exponent.

The requested value is 30.

Checks and common pitfalls: Use matching time units in the exponent.

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

06 / Foundation#Your turn

An exponential amount grows from 3 to 24 with doubling every 2 hours. How many hours elapsed?

  • Use the domain, units and sampling assumptions stated in the question.
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
First remove the initial amount.
Worked solution

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

  2. Calculate or simplify this relation.

    32t/2=24⇒2t/2=8⇒t=632^{t/2}=24\Rightarrow2^{t/2}=8\Rightarrow t=6

  3. Three doublings require six hours.

The requested value is 6.

Checks and common pitfalls: Three doublings require six hours.

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

07 / Foundation#Your turn

Find the base-10 logarithmic increase when a positive quantity is multiplied by 1000.

log⁡10(1000x)−log⁡10x\log_{10}(1000x)-\log_{10}x
  • Use the domain, units and sampling assumptions stated in the question.
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
A ratio becomes a logarithmic difference.
Worked solution

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

  2. Calculate or simplify this relation.

    log⁡10(1000x/x)=log⁡101000=3\log_{10}(1000x/x)=\log_{10}1000=3

  3. The result is independent of the initial positive x.

The requested value is 3.

Checks and common pitfalls: The result is independent of the initial positive x.

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

08 / Standard#Your turn

Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique.

f(x)=2x−3f(x)=2^x-3
  • Use the domain, units and sampling assumptions stated in the question.
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
A sign change needs continuity for a zero guarantee.
Worked solution

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

  2. Calculate or simplify this relation.

    f(1)=−1<0,f(2)=1>0f(1)=-1<0,\quad f(2)=1>0

  3. A sign change alone does not establish uniqueness.

Continuity gives existence; strict increase gives uniqueness.

Checks and common pitfalls: A sign change alone does not establish uniqueness.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

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

  • Use the domain, units and sampling assumptions stated in the question.
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

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

  2. Calculate or simplify this relation.

    L(t+1)−L(t)=3;E(t+1)/E(t)=2L(t+1)-L(t)=3;\quad E(t+1)/E(t)=2

  3. 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
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

10 / Standard#Your turn

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

  • Use the domain, units and sampling assumptions stated in the question.
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

  1. Distinguish the model output from the measured quantity.

  2. The model is continuous but the actual count is discrete; a decimal prediction is not an observation of a fractional individual.

  3. 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
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

A population is 400 and increases by 10% each year. Find it after two years.

  • Use the domain, units and sampling assumptions stated in the question.
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
The second percentage applies to the enlarged population.
Worked solution

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

  2. Calculate or simplify this relation.

    400(1.1)2=484400(1.1)^2=484

  3. This is compounding, not a single 20% increase.

The requested value is 484.

Checks and common pitfalls: This is compounding, not a single 20% increase.

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

12 / Transfer#Your turn

A sample starts at 320 mg and halves every 3 days. Find its mass after 9 days.

  • Use the domain, units and sampling assumptions stated in the question.
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
Nine days contains three half-lives.
Worked solution

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

  2. Calculate or simplify this relation.

    320(1/2)9/3=40320(1/2)^{9/3}=40

  3. Use matching time units in the exponent.

The requested value is 40.

Checks and common pitfalls: Use matching time units in the exponent.

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

13 / Transfer#Your turn

An exponential amount grows from 4 to 32 with doubling every 2 hours. How many hours elapsed?

  • Use the domain, units and sampling assumptions stated in the question.
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
First remove the initial amount.
Worked solution

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

  2. Calculate or simplify this relation.

    42t/2=32⇒2t/2=8⇒t=642^{t/2}=32\Rightarrow2^{t/2}=8\Rightarrow t=6

  3. Three doublings require six hours.

The requested value is 6.

Checks and common pitfalls: Three doublings require six hours.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for applications of functions ii?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • 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.

    Anticipated thinking

    • 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.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗