Kenneth ChengMathematical modeling
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Applications of functions I

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

高一必修 第一册(A版).pdf · 3.4 · PDF 100 / printed page 93

Revisit first: Power functions

TOPIC 01

Applications of functions I

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

What you will be able to explain

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

Modelling

State variables, units, assumptions and feasible inputs.

Validation

A fitted formula needs contextual checks and further 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 i 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.

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

Quadratic 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 taxi charges 2 units initially and 3 per kilometre. Find the charge for 4 km.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Add the fixed charge and the variable charge.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    C=2+3(4)=14C=2+3(4)=14

  3. The model assumes no extra surcharges or rounding.

The requested value is 14.

Checks and common pitfalls: The model assumes no extra surcharges or rounding.

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

02 / Standard#Worked example

Find the maximum profit under the continuous model.

P(q)=−(q−5)2+25,q≥0P(q)=-(q-5)^2+25,\quad q\ge0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
The squared term is nonnegative.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    P(q)≤25;q=5P(q)\le25;\quad q=5

  3. The maximizing output is nonnegative and therefore feasible.

The requested value is 25.

Checks and common pitfalls: The maximizing output is nonnegative and therefore feasible.

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

03 / Transfer#Worked example

A learner fits a straight line through two observations and claims the model is valid forever. Evaluate the claim.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Distinguish fitting from validation.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Check further observations and the mechanism generating the data; state a justified range of use.

  3. A model is conditional on assumptions and evidence.

Two observations determine a line but do not validate unlimited extrapolation.

Checks and common pitfalls: A model is conditional on assumptions and evidence.

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 taxi charges 3 units initially and 3 per kilometre. Find the charge for 5 km.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Add the fixed charge and the variable charge.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    C=3+3(5)=18C=3+3(5)=18

  3. The model assumes no extra surcharges or rounding.

The requested value is 18.

Checks and common pitfalls: The model assumes no extra surcharges or rounding.

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

05 / Foundation#Your turn

Find the maximum profit under the continuous model.

P(q)=−(q−6)2+36,q≥0P(q)=-(q-6)^2+36,\quad q\ge0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
The squared term is nonnegative.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    P(q)≤36;q=6P(q)\le36;\quad q=6

  3. The maximizing output is nonnegative and therefore feasible.

The requested value is 36.

Checks and common pitfalls: The maximizing output is nonnegative and therefore feasible.

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

06 / Foundation#Your turn

A journey covers equal distances at speeds 6 and 18. Find its average speed.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Average speed is total distance divided by total time.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    v=2dd/6+d/18=9v=\frac{2d}{d/6+d/18}=9

  3. Equal distances do not imply equal travel times.

The requested value is 9.

Checks and common pitfalls: Equal distances do not imply equal travel times.

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

07 / Foundation#Your turn

A service costs C=3+4n for n whole items. A bill is 23. Find n.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Subtract the fixed charge before dividing.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    4n=(23)−3=20⇒n=54n=(23)-3=20\Rightarrow n=5

  3. The answer must be a nonnegative integer.

The requested value is 5.

Checks and common pitfalls: The answer must be a nonnegative integer.

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

08 / Standard#Your turn

A budget is 17 and each item costs 5. Find the greatest number of whole items affordable.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Round the continuous upper bound down, not to the nearest integer.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    5n≤17;n∈Z≥0;nmax⁡=35n\le17;\quad n\in\mathbb Z_{\ge0};\quad n_{\max}=3

  3. A rounded-up purchase would exceed the budget.

The requested value is 3.

Checks and common pitfalls: A rounded-up purchase would exceed the budget.

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

09 / Standard#Your turn

Why should the input domain of a ticket-revenue model be discrete and bounded?

R(n)=pnR(n)=pn
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
State what one unit of the input represents.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    n∈{0,1,…,N}n\in\{0,1,\ldots,N\}

  3. A smooth graph is an aid; fractional tickets and over-capacity sales may be infeasible.

Ticket counts are integers from zero to capacity.

Checks and common pitfalls: A smooth graph is an aid; fractional tickets and over-capacity sales may be infeasible.

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 fare is 8 for 0≤d≤2, then 8+3(d−2) for d>2. Find the fare at d=5 and explain why the branches agree at d=2.

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

Working and explanation

BUILD THE REASONING

Hint 1
Choose the branch containing the input.
Hint 2
The added distance is measured beyond two, not from zero.
Worked solution

  1. Choose the branch containing the input.

  2. Calculate or simplify this relation.

    C(5)=8+3(5−2)=17;8+3(2−2)=8C(5)=8+3(5-2)=17;\quad8+3(2-2)=8

  3. The two formulas give the same boundary value, so the fare has no jump there.

The requested value is 17.

Checks and common pitfalls: The two formulas give the same boundary value, so the fare has no jump there.

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

11 / Standard#Your turn

A taxi charges 4 units initially and 3 per kilometre. Find the charge for 6 km.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Add the fixed charge and the variable charge.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    C=4+3(6)=22C=4+3(6)=22

  3. The model assumes no extra surcharges or rounding.

The requested value is 22.

Checks and common pitfalls: The model assumes no extra surcharges or rounding.

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

12 / Transfer#Your turn

Find the maximum profit under the continuous model.

P(q)=−(q−7)2+49,q≥0P(q)=-(q-7)^2+49,\quad q\ge0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
The squared term is nonnegative.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    P(q)≤49;q=7P(q)\le49;\quad q=7

  3. The maximizing output is nonnegative and therefore feasible.

The requested value is 49.

Checks and common pitfalls: The maximizing output is nonnegative and therefore feasible.

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

13 / Transfer#Your turn

A journey covers equal distances at speeds 8 and 24. Find its average speed.

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

Working and explanation

BUILD THE REASONING

Hint 1
Define the variable and its units, form a model, and check feasible inputs.
Hint 2
Average speed is total distance divided by total time.
Worked solution

  1. Define the variable and its units, form a model, and check feasible inputs.

  2. Calculate or simplify this relation.

    v=2dd/8+d/24=12v=\frac{2d}{d/8+d/24}=12

  3. Equal distances do not imply equal travel times.

The requested value is 12.

Checks and common pitfalls: Equal distances do not imply equal travel times.

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 i?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Interpret and use applications of functions i with domain checks.
    • Modelling: State variables, units, assumptions and feasible inputs.
    • Validation: A fitted formula needs contextual checks and further evidence.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: State variables, units, assumptions and feasible inputs.
    • Expected reasoning: A fitted formula needs contextual checks and further evidence.
    • Expected correction: Average speed is not generally the average of two speeds.

    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 ↗