LEARN · EXPLAIN · REVISE

Find the normal to y=x² at x=t.

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

TOPIC 01

Derivative: concept and meaning

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

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

01 / Transfer#Worked example

t=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.

m_{\rm normal}=-1/f\prime(t)

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$m_{\rm tangent}=8\ne0$
Apply the stated relation and retain its conditions.
$m_{\rm normal}=-1/8$
The given derivative is nonzero; a zero tangent slope would require a vertical normal.

The requested relation or conclusion is shown below.

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

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

Teacher preparation and assessment

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