LEARN · EXPLAIN · REVISE

A tank volume V(h)=3h². Find dV/dh at h=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 / Standard#Your turn

t=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=6h

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$V'(11)=66$
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.

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.