Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Find the tangent to y=x² at x=t.

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

Return to the lesson / paper ↗

TOPIC 01

Derivatives and their applications: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

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

Find the tangent to y=x² at x=t.

t=23t=23
  • 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=46m=46

  3. Apply the stated relation and retain its conditions.

    y−529=46(x−23)y-529=46(x-23)

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

The requested relation or conclusion is shown below.

y−529=46(x−23)y-529=46(x-23)

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.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗