Twenty minutes or less.

5-minute version

Three sub-topics. One sentence each.

• Critical points — solve $f^{\prime }(x)=0$, classify with $f^{\prime \prime }$ or sign-change.
• Optimization — write the objective in one variable, then minimise/maximise.
• Related rates — differentiate the geometric equation w.r.t. $t$, chain rule.

Open the cheatsheet quiz, do 3 easy questions, close it. You’re prepped.

20-minute prep plan

Time	Action
0–5 min	Skim the cheatsheet tables.
5–10 min	Do one worked example from each PDF — covering pen, write it out.
10–15 min	Take the cheatsheet quiz. Don’t worry about the score.
15–20 min	Read the matching “common mistakes” + worked example in the in-depth note.

What to revise first

Most students slip on two specific things in this week:

1. Differentiating before substituting in related-rates. If your $dA/dt$ comes out to a number with no $dr/dt$ in it, you skipped a step.
2. Forgetting to use the constraint in optimisation. If you’re staring at two variables, you haven’t substituted yet.

Key formulas

$$f^{\prime }(x)=0\text{(critical point)}{f}^{\prime \prime }(c)>0\Rightarrow \text{min}{f}^{\prime \prime }(c)<0\Rightarrow \text{max}$$ $$\text{Optimise:}\text{objective}(x)\stackrel{d/dx}{\to }0\stackrel{\text{classify}}{\to }\text{answer}$$ $$\text{Related rate:}Q=g(P)\stackrel{d/dt}{\to }\dot{Q}=g^{\prime }(P)\dot{P}$$

Likely workshop tasks

Task type	What the setup usually looks like
Critical points	Differentiate, solve $f^{\prime }(x)=0$, then classify
Optimization	Write a constraint and an objective, reduce to one variable, then optimize
Related rates	Start from a geometry formula and differentiate both sides with respect to $t$

Mistakes to avoid

• Treating $f^{\prime \prime }(c)=0$ as a conclusion. It isn’t.
• Optimising the wrong quantity (read the question twice).
• Plugging numbers in before differentiating in related rates.
• Forgetting endpoint checks on closed-interval problems.
• Sign / unit errors in the final answer.

Mini self-test

Try these without notes. Five minutes total.

1. Find and classify the critical points of $f(x)=x^2\ln (x)$ for $x>0$.
2. Find two positive numbers with sum 100 such that the product of one and the square of the other is maximum.
3. The radius of a circle grows at $2$ cm/s. How fast is the area growing when $r=10$ cm?

Answers:

Question	Answer
1	Minimum at $\left(\frac{1}{\sqrt{e}},-\frac{1}{2e}\right)$
2	$33\frac{1}{3}$ and $66\frac{2}{3}$; maximum product $=148148.15$
3	$40\pi {\text{cm}}^2/\text{s}$

Done checklist

• Read the cheatsheet tables.
• One worked example from each PDF, copied out longhand.
• Cheatsheet quiz attempted.
• Mini self-test attempted.

That’s it. Close the laptop.

Source files used

• school-stuff/EGD105-Calculus/Week10/Finding Critical Points.pdf
• school-stuff/EGD105-Calculus/Week10/Optimization.pdf
• school-stuff/EGD105-Calculus/Week10/Related Rates of Change.pdf