Kenneth ChengMathematical modeling
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Derivative: concept and meaning

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

高二選擇性必修 第二册(A版).pdf · 5.1 · PDF 64 / printed page 59

Revisit first: Function concept and representations

TOPIC 01

Derivative: concept and meaning

Distinguish average and instantaneous rates and derive tangents from a limit.

What you will be able to explain

  • Distinguish average and instantaneous rates and derive tangents from a limit.
  • Justify the method and check the conditions in a new situation.

Defining relation

Distinguish average and instantaneous rates and derive tangents from a limit.

f′(a)=lim⁡h→0[f(a+h)−f(a)]/hf\prime(a)=\lim_{h\to0}[f(a+h)-f(a)]/h

Conditions

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does a graph without a jump always have a tangent slope?

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 a smooth point with a corner by left and right quotients.

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 derivative of x² at x=t from the limit.

t=2t=2
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
[(t+h)2−t2]/h=2t+h[(t+h)^2-t^2]/h=2t+h
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(2)=lim⁡h→0(4+h)=4f'(2)=\lim_{h\to0}(4+h)=4

  3. Cancel h only while h≠0, then take the limit.

The requested value is 4.

Checks and common pitfalls: Cancel h only while h≠0, then take the limit.

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

02 / Standard#Worked example

Is |x−t| differentiable at x=t?

t=3t=3
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Compare the left and right difference quotients.
Hint 2
Use this intermediate relation.
∣h∣/h|h|/h
Worked solution

  1. Compare the left and right difference quotients.

  2. Apply the stated relation and retain its conditions.

    h>0:∣h∣/h=1h>0: |h|/h=1

  3. Apply the stated relation and retain its conditions.

    h<0:∣h∣/h=−1h<0: |h|/h=−1

  4. Continuity does not remove the mismatch in the one-sided derivatives.

The requested relation or conclusion is shown below.

f′(3) undefinedf\prime(3)\text{ undefined}

Checks and common pitfalls: Continuity does not remove the mismatch in the one-sided derivatives.

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.

03 / Transfer#Worked example

Find the normal to y=x² at x=t.

t=4t=4
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
mnormal=−1/f′(t)m_{\rm normal}=-1/f\prime(t)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    mtangent=8≠0m_{\rm tangent}=8\ne0

  3. Apply the stated relation and retain its conditions.

    mnormal=−1/8m_{\rm normal}=-1/8

  4. The given derivative is nonzero; a zero tangent slope would require a vertical normal.

The requested relation or conclusion is shown below.

y−16=−18(x−4)y-16=-\frac1{8}(x-4)

Checks and common pitfalls: The given derivative is nonzero; a zero tangent slope would require a vertical normal.

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.

04 / Foundation#Your turn

Find the derivative of x² at x=t from the limit.

t=5t=5
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
[(t+h)2−t2]/h=2t+h[(t+h)^2-t^2]/h=2t+h
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(5)=lim⁡h→0(10+h)=10f'(5)=\lim_{h\to0}(10+h)=10

  3. Cancel h only while h≠0, then take the limit.

The requested value is 10.

Checks and common pitfalls: Cancel h only while h≠0, then take the limit.

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

05 / Foundation#Your turn

Find the average rate of x² from t to t+1.

t=6t=6
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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(t+1)−f(t)]/1[f(t+1)-f(t)]/1
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    (7)2−62=13(7)^2-6^2=13

  3. A secant rate over a finite interval is not generally the endpoint derivative.

The requested value is 13.

Checks and common pitfalls: A secant rate over a finite interval is not generally the endpoint derivative.

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

06 / Foundation#Your turn

Position is s(u)=u²+3u. Find velocity at u=t.

t=7t=7
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
v(u)=s′(u)=2u+3v(u)=s\prime(u)=2u+3
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    v(7)=17v(7)=17

  3. Velocity has position-units per time-unit; acceleration would require another derivative.

The requested value is 17.

Checks and common pitfalls: Velocity has position-units per time-unit; acceleration would require another derivative.

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

07 / Foundation#Your turn

Find the tangent to y=x² at x=t.

t=8t=8
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
m=f′(t),P=(t,t2)m=f\prime(t),\quad P=(t,t^2)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    m=16m=16

  3. Apply the stated relation and retain its conditions.

    y−64=16(x−8)y-64=16(x-8)

  4. A slope alone is not a line equation; include the point of contact.

The requested relation or conclusion is shown below.

y−64=16(x−8)y-64=16(x-8)

Checks and common pitfalls: A slope alone is not a line equation; include the point of contact.

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

Find the tangent to y=x² at x=t.

t=9t=9
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
m=f′(t),P=(t,t2)m=f\prime(t),\quad P=(t,t^2)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    m=18m=18

  3. Apply the stated relation and retain its conditions.

    y−81=18(x−9)y-81=18(x-9)

  4. A slope alone is not a line equation; include the point of contact.

The requested relation or conclusion is shown below.

y−81=18(x−9)y-81=18(x-9)

Checks and common pitfalls: A slope alone is not a line equation; include the point of contact.

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

Is |x−t| differentiable at x=t?

t=10t=10
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Compare the left and right difference quotients.
Hint 2
Use this intermediate relation.
∣h∣/h|h|/h
Worked solution

  1. Compare the left and right difference quotients.

  2. Apply the stated relation and retain its conditions.

    h>0:∣h∣/h=1h>0: |h|/h=1

  3. Apply the stated relation and retain its conditions.

    h<0:∣h∣/h=−1h<0: |h|/h=−1

  4. Continuity does not remove the mismatch in the one-sided derivatives.

The requested relation or conclusion is shown below.

f′(10) undefinedf\prime(10)\text{ undefined}

Checks and common pitfalls: Continuity does not remove the mismatch in the one-sided derivatives.

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.

10 / Standard#Your turn

A tank volume V(h)=3h². Find dV/dh at h=t.

t=11t=11
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
dV/dh=6hdV/dh=6h
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    V′(11)=66V'(11)=66

  3. The derivative is volume change per height change, not necessarily volume change per time.

The requested value is 66.

Checks and common pitfalls: The derivative is volume change per height change, not necessarily volume change per time.

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

11 / Standard#Your turn

Find the normal to y=x² at x=t.

t=12t=12
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
mnormal=−1/f′(t)m_{\rm normal}=-1/f\prime(t)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    mtangent=24≠0m_{\rm tangent}=24\ne0

  3. Apply the stated relation and retain its conditions.

    mnormal=−1/24m_{\rm normal}=-1/24

  4. The given derivative is nonzero; a zero tangent slope would require a vertical normal.

The requested relation or conclusion is shown below.

y−144=−124(x−12)y-144=-\frac1{24}(x-12)

Checks and common pitfalls: The given derivative is nonzero; a zero tangent slope would require a vertical normal.

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.

12 / Transfer#Your turn

A tank volume V(h)=3h². Find dV/dh at h=t.

t=13t=13
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
dV/dh=6hdV/dh=6h
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    V′(13)=78V'(13)=78

  3. The derivative is volume change per height change, not necessarily volume change per time.

The requested value is 78.

Checks and common pitfalls: The derivative is volume change per height change, not necessarily volume change per time.

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

13 / Transfer#Your turn

Find the normal to y=x² at x=t.

t=14t=14
  • The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.
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.
mnormal=−1/f′(t)m_{\rm normal}=-1/f\prime(t)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    mtangent=28≠0m_{\rm tangent}=28\ne0

  3. Apply the stated relation and retain its conditions.

    mnormal=−1/28m_{\rm normal}=-1/28

  4. The given derivative is nonzero; a zero tangent slope would require a vertical normal.

The requested relation or conclusion is shown below.

y−196=−128(x−14)y-196=-\frac1{28}(x-14)

Checks and common pitfalls: The given derivative is nonzero; a zero tangent slope would require a vertical normal.

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.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Distinguish average and instantaneous rates and derive tangents from a limit.
    • Which condition is essential in derivative: concept and meaning?
    • Does a graph without a jump always have a tangent slope?

    Board plan

    • Defining relation: Distinguish average and instantaneous rates and derive tangents from a limit.
      f′(a)=lim⁡h→0[f(a+h)−f(a)]/hf\prime(a)=\lim_{h\to0}[f(a+h)-f(a)]/h
    • Conditions: The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

    Anticipated thinking

    • Continuity alone does not imply differentiability.

    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 ↗