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Applications of derivatives — Formula Sheet

3 formulas from Calculus I, each with what it means, when to use it and the trap it sets.

Critical points and concavity

f^{'} (x) = 0 or undefined; f^{''} (x) > 0 \Rightarrow minimum

What it says: The first derivative finds the flat spots; the second says whether each is a peak or a valley.
When to reach for it: Sketching a curve, or classifying a stationary point.
Watch out: f''(x) = 0 decides nothing on its own — it can be an inflection. Fall back to a sign chart of f'.

f (x) = x^{3} - 3 x \Rightarrow f^{'} (x) = 3 x^{2} - 3 = 0 \Rightarrow x = \pm 1

f''(1) = 6 > 0, so x = 1 is a minimum.

Optimisation workflow

constraint \to one variable \to f^{'} (x) = 0 \to check endpoints

What it says: Use the constraint to remove a variable, then find the stationary point of what is left.
When to reach for it: Largest area, cheapest material, shortest distance — any "maximise or minimise" word problem.
Watch out: On a closed interval the maximum can sit at an endpoint where the derivative is not zero. Always test the ends as well.

P = 2 x + 2 y = 40, A = x y \Rightarrow A (x) = x (20 - x)

The constraint turned two variables into one before differentiating.

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