Application of Derivatives Class 12: notes and important questions

Application of Derivatives — revision notes

1. Rate of change

\(\dfrac{dy}{dx}\) is the rate of change of \(y\) with respect to \(x\). If both depend on time, \(\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}\). Write a relation between the variables first, differentiate with respect to \(t\), and only then substitute the instantaneous values. In economics, marginal cost \(= C'(x)\) and marginal revenue \(= R'(x)\).

2. Increasing and decreasing functions

• \(f' \gt 0\) on an open interval \(\Rightarrow f\) strictly increasing there; \(f' \lt 0 \Rightarrow\) strictly decreasing.
• \(f' \ge 0\) with \(f' = 0\) only at isolated points still gives strictly increasing (e.g. \(x^3\)).
• Method: find where \(f' = 0\) or \(f'\) is undefined, split the line, test the sign of \(f'\) in each part. Square factors such as \((x - 1)^2\) do not change sign.
• Inequalities: to show \(f(x) \gt g(x)\) for \(x \gt a\), prove \(h = f - g\) is increasing and \(h(a) \ge 0\).

3. Local maxima and minima

• Critical point: \(f'(c) = 0\) or \(f'(c)\) does not exist.
• First derivative test: \(f'\) changes \(+ \to -\): local max; \(- \to +\): local min; no change: neither.
• Second derivative test: \(f'(c) = 0\) and \(f''(c) \lt 0\): local max; \(f''(c) \gt 0\): local min; \(f''(c) = 0\): test fails, use the first derivative test.

4. Absolute extrema on \([a, b]\)

Evaluate \(f\) at all critical points in \((a, b)\) and at the end points \(a, b\); the largest value is the absolute maximum, the smallest the absolute minimum.

5. Optimisation strategy

1. Draw/label; write the quantity to optimise.
2. Use the constraint to reduce it to one variable; state the domain.
3. Differentiate, solve, and justify max/min (second derivative or sign change).
4. Answer the question actually asked (dimensions, cost, …) with units.

Worked example 1

A spherical balloon is inflated at \(100\ \text{cm}^3/\text{s}\). When \(r = 5\) cm: \(\dfrac{dV}{dt} = 4\pi r^2\dfrac{dr}{dt} \Rightarrow \dfrac{dr}{dt} = \dfrac{100}{100\pi} = \dfrac1\pi\ \text{cm/s}\).

Worked example 2

\(f(x) = 2x^3 - 3x^2 - 12x\): \(f'(x) = 6(x - 2)(x + 1)\). Increasing on \((-\infty, -1]\) and \([2, \infty)\), decreasing on \([-1, 2]\); local max \(f(-1) = 7\), local min \(f(2) = -20\).

Worked example 3

Of all rectangles with perimeter \(40\) cm, the area \(A = x(20 - x)\) is greatest when \(A'(x) = 20 - 2x = 0\), i.e. the square of side \(10\) cm (\(A'' = -2 \lt 0\)).

Common errors

• Substituting the particular value before differentiating in a rate problem.
• Forgetting end points when finding absolute extrema; confusing a local maximum with the absolute maximum.
• Ignoring points where \(f'\) is undefined (e.g. \(x^{2/3}\) at \(0\)).
• Not justifying that the stationary point is a maximum/minimum — this carries a mark.
• Rejecting a variable value that is outside the domain without comment (e.g. negative lengths): say why it is rejected.

Board tips

• Tangents and normals and approximations have been removed from the syllabus; questions come from rate of change, monotonicity and maxima/minima.
• Write intervals clearly; either open or closed end points are accepted for strict monotonicity where \(f\) is continuous.
• In case-based questions, read what each part asks — part (iii) often needs the value (profit, area), not just \(x\).