Kenneth ChengMathematical modeling
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Applications of derivatives

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

高二選擇性必修 第二册(A版).pdf · 5.3 · PDF 89 / printed page 84

Revisit first: Derivative operations

TOPIC 01

Applications of derivatives

Use derivative signs, boundary values and models to justify extrema and optimization.

What you will be able to explain

  • Use derivative signs, boundary values and models to justify extrema and optimization.
  • Justify the method and check the conditions in a new situation.

Defining relation

Use derivative signs, boundary values and models to justify extrema and optimization.

f′>0⇒f increasingf\prime>0\Rightarrow f\text{ increasing}

Conditions

A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can a stationary point fail to be an extremum?

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

y=ax²; tangent slope at x=t is 2; signed integral from 0 to t is 0.3333. Signed integral can be negative.

y=ax²; tangent slope at x=t is 2; signed integral from 0 to t is 0.3333. Signed integral can be negative.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare x² and x³ at zero and explain their derivative sign changes.

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

Find the minimizing x over all real x.

f(x)=x2−4x+1f(x)=x^2-4x+1
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=2x−2tf\prime(x)=2x-2t
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′(x)=0⇒x=2f'(x)=0\Rightarrow x=2

  3. Apply the stated relation and retain its conditions.

    f′′(x)=2>0f''(x)=2>0

  4. A positive quadratic has a unique global minimum.

The requested value is 2.

Checks and common pitfalls: A positive quadratic has a unique global minimum.

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

02 / Standard#Worked example

A rectangle has perimeter 4t. Find its maximum area.

t=3t=3
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
y=2t−x;A=x(2t−x)y=2t-x;\quad A=x(2t-x)
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    A′=6−2x=0⇒x=3A'=6-2x=0\Rightarrow x=3

  3. Apply the stated relation and retain its conditions.

    Amax⁡=9A_{\max}=9

  4. The domain 0<x<2t admits the square, and boundary areas approach zero.

The requested value is 9.

Checks and common pitfalls: The domain 0<x<2t admits the square, and boundary areas approach zero.

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

03 / Transfer#Worked example

Profit is P(x)=−x²+2tx−3, for 0≤x≤t+2. Find maximum profit.

t=4t=4
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
P′(x)=2t−2xP\prime(x)=2t-2x
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    x=4 lies in the domainx=4\text{ lies in the domain}

  3. Apply the stated relation and retain its conditions.

    P(4)=13P(4)=13

  4. Apply the stated relation and retain its conditions.

    P′′=−2<0P''=-2<0

  5. The feasible vertex gives the optimum; the model concerns profit, not gross revenue.

The requested value is 13.

Checks and common pitfalls: The feasible vertex gives the optimum; the model concerns profit, not gross revenue.

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

04 / Foundation#Your turn

Find the minimizing x over all real x.

f(x)=x2−10x+1f(x)=x^2-10x+1
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=2x−2tf\prime(x)=2x-2t
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′(x)=0⇒x=5f'(x)=0\Rightarrow x=5

  3. Apply the stated relation and retain its conditions.

    f′′(x)=2>0f''(x)=2>0

  4. A positive quadratic has a unique global minimum.

The requested value is 5.

Checks and common pitfalls: A positive quadratic has a unique global minimum.

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

05 / Foundation#Your turn

Find the maximum of f on [0,t].

f(x)=x2,t=6f(x)=x^2,\quad t=6
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=2x≥0f\prime(x)=2x\ge0
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f(0)=0,f(6)=36f(0)=0,\quad f(6)=36

  3. Compare the endpoints of the closed interval.

The requested value is 36.

Checks and common pitfalls: Compare the endpoints of the closed interval.

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

06 / Foundation#Your turn

Find the increasing and decreasing intervals.

f(x)=x2−14xf(x)=x^2-14x
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=2(x−t)f\prime(x)=2(x-t)
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    x<7:f′<0;x>7:f′>0x<7:f'<0;\quad x>7:f'>0

  3. The derivative sign changes from negative to positive.

The requested relation or conclusion is shown below.

(−∞,7):↘;(7,∞):↗(-\infty,7):\searrow;\quad(7,\infty):\nearrow

Checks and common pitfalls: The derivative sign changes from negative to positive.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

07 / Foundation#Your turn

Is x=t a local extremum of (x−t)³?

t=8t=8
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=3(x−t)2f\prime(x)=3(x-t)^2
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′(x)>0(x≠8)f\prime(x)>0\quad(x\ne8)

  3. The derivative has the same sign on both sides.

The requested relation or conclusion is shown below.

stationary inflection; no local extremum\text{stationary inflection; no local extremum}

Checks and common pitfalls: The derivative has the same sign on both sides.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

08 / Standard#Your turn

Is x=t a local extremum of (x−t)³?

t=9t=9
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=3(x−t)2f\prime(x)=3(x-t)^2
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′(x)>0(x≠9)f\prime(x)>0\quad(x\ne9)

  3. The derivative has the same sign on both sides.

The requested relation or conclusion is shown below.

stationary inflection; no local extremum\text{stationary inflection; no local extremum}

Checks and common pitfalls: The derivative has the same sign on both sides.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

09 / Standard#Your turn

A rectangle has perimeter 4t. Find its maximum area.

t=10t=10
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
y=2t−x;A=x(2t−x)y=2t-x;\quad A=x(2t-x)
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    A′=20−2x=0⇒x=10A'=20-2x=0\Rightarrow x=10

  3. Apply the stated relation and retain its conditions.

    Amax⁡=100A_{\max}=100

  4. The domain 0<x<2t admits the square, and boundary areas approach zero.

The requested value is 100.

Checks and common pitfalls: The domain 0<x<2t admits the square, and boundary areas approach zero.

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

10 / Standard#Your turn

Find the minimum of x+t²/x for x>0.

t=11t=11
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=1−t2/x2f\prime(x)=1-t^2/x^2
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′=0⇒x=11f'=0\Rightarrow x=11

  3. Apply the stated relation and retain its conditions.

    f′′=242/x3>0f''=242/x^3>0

  4. Apply the stated relation and retain its conditions.

    f(11)=22f(11)=22

  5. The negative critical solution is outside the positive domain.

The requested value is 22.

Checks and common pitfalls: The negative critical solution is outside the positive domain.

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

11 / Standard#Your turn

Profit is P(x)=−x²+2tx−3, for 0≤x≤t+2. Find maximum profit.

t=12t=12
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
P′(x)=2t−2xP\prime(x)=2t-2x
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    x=12 lies in the domainx=12\text{ lies in the domain}

  3. Apply the stated relation and retain its conditions.

    P(12)=141P(12)=141

  4. Apply the stated relation and retain its conditions.

    P′′=−2<0P''=-2<0

  5. The feasible vertex gives the optimum; the model concerns profit, not gross revenue.

The requested value is 141.

Checks and common pitfalls: The feasible vertex gives the optimum; the model concerns profit, not gross revenue.

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

12 / Transfer#Your turn

Find the minimum of x+t²/x for x>0.

t=13t=13
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′(x)=1−t2/x2f\prime(x)=1-t^2/x^2
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′=0⇒x=13f'=0\Rightarrow x=13

  3. Apply the stated relation and retain its conditions.

    f′′=338/x3>0f''=338/x^3>0

  4. Apply the stated relation and retain its conditions.

    f(13)=26f(13)=26

  5. The negative critical solution is outside the positive domain.

The requested value is 26.

Checks and common pitfalls: The negative critical solution is outside the positive domain.

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

13 / Transfer#Your turn

Profit is P(x)=−x²+2tx−3, for 0≤x≤t+2. Find maximum profit.

t=14t=14
  • A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
P′(x)=2t−2xP\prime(x)=2t-2x
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    x=14 lies in the domainx=14\text{ lies in the domain}

  3. Apply the stated relation and retain its conditions.

    P(14)=193P(14)=193

  4. Apply the stated relation and retain its conditions.

    P′′=−2<0P''=-2<0

  5. The feasible vertex gives the optimum; the model concerns profit, not gross revenue.

The requested value is 193.

Checks and common pitfalls: The feasible vertex gives the optimum; the model concerns profit, not gross revenue.

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

Focus on one question

End-of-lesson check and correction

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    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • Use derivative signs, boundary values and models to justify extrema and optimization.
    • Which condition is essential in applications of derivatives?
    • Can a stationary point fail to be an extremum?

    Board plan

    • Defining relation: Use derivative signs, boundary values and models to justify extrema and optimization.
      f′>0⇒f increasingf\prime>0\Rightarrow f\text{ increasing}
    • Conditions: A critical point needs a sign or second-derivative check; closed-interval global extrema also require endpoints.

    Anticipated thinking

    • A zero derivative alone does not imply a maximum or minimum.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗