# Gradient, tangent plane, linearization, and directional derivative

> ML preparation item.

- Stable ID: `mit-18-02sc-f2010-exam2-q1`
- Area: Mathematics
- Topic: Calculus
- Origin: external
- Expected time: 12 minutes
- Skills: gradient, tangent-plane, linear-approximation, directional-derivative
- Item JSON: https://mlprep.iwase.dev/data/items/mit-18-02sc-f2010-exam2-q1.json

This site does not reproduce the third-party problem.

- Source: MIT 18.02SC Multivariable Calculus
- Document: Exam 2
- Locator: Question 1(a–d), page 1
- Official answer PDF page(s): 1
- Original problem: https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/resources/mit18_02sc_exam2/
- Official answer: https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/resources/mit18_02sc_exam2sol/
- Answer status: official-solution


## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `mit-18-02sc-f2010-exam2-q1`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.