LEARN · EXPLAIN · REVISE

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

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 / Foundation#Your turn

t=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

Worked solution

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.

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.