Week 10 — Applications of Differentiation

What this week is about

This is the applied half of differentiation. You already know how to take a derivative — this week you use it to answer three useful questions:

1. Where does a function peak or bottom out? → Critical points + first/second derivative tests.
2. What’s the best size/shape/cost? → Optimization: build a function, find the extremum.
3. If one quantity changes, how fast does another change with it? → Related rates, via the chain rule.

All three are one technique — $f^{\prime }(x)=0$ — applied to different questions. See the in-depth note for the connection.

Notes in this week

• Cheatsheet — every rule, table, and recipe in one scannable page. Includes the quiz (mixed difficulty, reshuffles every visit).
• In-depth analysis — why each technique works, full worked example per topic, the Taylor-series argument behind the second derivative test, the exam-style template.

Any notes you add to this folder will appear here automatically.

What I need to know before the workshop

• Power rule, product rule, quotient rule, chain rule
• How to solve $f^{\prime }(x)=0$ for common function shapes
• What the sign of $f^{\prime \prime }(x)$ tells you about a critical point
• How to differentiate both sides of an equation with respect to $t$

Assessment relevance

• Exam questions in EGD105 lean heavily on these three patterns.
• Optimization problems show up almost every paper.