High yield5 mapped NDA questions
Build the idea
Applications convert derivatives into geometry and optimisation. A stationary point is only a candidate: use sign change or the second derivative to classify it. In word problems, define the quantity to optimise in one variable before differentiating.
Concept
Derivatives are applied to find tangents and normals, to determine where a function increases or decreases, and to locate maxima and minima. Optimisation ('maximum area/volume') problems are a classic format.
At a maximum or minimum the first derivative is zero.
Key Points & Formulas
- Slope of tangent at a point = value of dy/dx there.
- f'(x) > 0 ⇒ increasing; f'(x) < 0 ⇒ decreasing.
- Critical points: solve f'(x) = 0.
- Second-derivative test: f''<0 ⇒ maximum; f''>0 ⇒ minimum.
Worked Example
Q. Find the local maximum value of f(x) = x³ − 12x.
f'(x) = 3x² − 12 = 0 ⇒ x = ±2.
f''(x) = 6x; at x = −2, f'' = −12 < 0 (maximum).
Maximum value f(−2) = −8 + 24 = 16.