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For y=x² at x=3, what is the tangent slope?

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

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

TOPIC 01

Rates and tangent meaning

Derive tangent slope from shrinking secants, then connect slope, point and local approximation.

What you will be able to explain

  • Derive tangent slope from shrinking secants, then connect slope, point and local approximation.
  • Distinguish average rate, instantaneous rate, height, normal and inverse rate.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

For y=x² at x=3, what is the tangent slope?

y=x2; a=3y=x^2;\ a=3
  • Use the stated real domain and distinguish stationary candidates, attained extrema and actual concavity changes.
Skills and prerequisite lessons

Scope: JM02 · This task uses elementary probability rules or polynomial calculus named in the JAE syllabus.

  1. 9
  2. 6
  3. 3
  4. 0
Working and explanation
BUILD THE REASONING
Hint 1
Differentiate the function, then evaluate its derivative.
Hint 2
Start from this intermediate relation.
f′(x)=2xf'(x)=2x
Worked solution
  1. Differentiate the function, then evaluate its derivative.
  2. Establish and simplify this mathematical relation.
    f′(x)=2xf'(x)=2x
  3. Establish and simplify this mathematical relation.
    f′(3)=6f'(3)=6
  4. The height is 9 but the slope is 6.
6
Checks and common pitfalls: The height is 9 but the slope is 6.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5PredictPredict slopes of secants to x² at a=1 from both sides.
8Build the ideaSample response: quotient 2+h tends to 2; h cannot initially be zero.
10Guided comparisonSample response: slope 4 at (2,4) gives y−4=4(x−2); normal slope is −1/4.
12Independent practice and discussionContrast the corner task with normal and local-error tasks; require a reason before calculation.
5Exit and next stepExit: explain why f(3)=9 but f′(3)=6, and when a normal is vertical.
Question sequence
  • Predict slopes of secants to x² at a=1 from both sides.
  • Sample response: slope 4 at (2,4) gives y−4=4(x−2); normal slope is −1/4.
Board development
  • From secant to tangent
    f′(a)=lim⁡h→0[f(a+h)−f(a)]/hf\prime(a)=\lim_{h\to0}[f(a+h)-f(a)]/h
Expected correct and incorrect responses
  • Correct sample: 6
  • Anticipated wrong response: 9 — Substitutes the function height for its derivative.
  • Anticipated wrong response: Derivative is 0. — Applies a differentiable interior-minimum theorem to a corner.
  • Confusing f(a) with f′(a).
  • Treating a corner as a horizontal tangent.
Follow-up questions
  • Follow-up: Differentiate the function, then evaluate its derivative. Then explain why 9 fails.
  • Follow-up: Compare left and right difference quotients. Then explain why Derivative is 0. fails.
  • Relate the vertical gap h² to the accuracy of the local approximation.
Remediation
  • Sample response: quotient 2+h tends to 2; h cannot initially be zero.
  • Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.
Exit check
  • Exit: explain why f(3)=9 but f′(3)=6, and when a normal is vertical.
Assessment checklist and sample responses
  • Representation and conditions
    For h≠0, [(1+h)²−1]/h=2+h; its limit is 2.
  • Connected mathematical reasoning
    At (2,4), f′(2)=4, so y−4=4(x−2).
  • Interpretation and boundary check
    The exact minus predicted value is h²≥0, even for negative h.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗