Differentiation (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
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Differentiation (AS Unit 1: Pure Mathematics A) explained
The sign of the first derivative dy/dx determines whether f(x) is increasing or decreasing.
Read the full explanation
On an interval, if f'(x) ≥ 0 throughout, then f is increasing; if f'(x) > 0 throughout, then f is strictly increasing. Similarly, f'(x) ≤ 0 gives decreasing and f'(x) < 0 gives strictly decreasing. These are sufficient conditions, not necessary ones: f(x) = x³ is strictly increasing on [-1, 1] even though f'(0) = 0, so equality may occur at isolated points without destroying monotonicity. To find intervals, differentiate f(x), solve f'(x) ≥ 0 or f'(x) ≤ 0, factorise where possible, and analyse the sign of f'(x) across critical boundary points.
Your focus
- Define increasing and decreasing functions in terms of the sign of the first derivative.
- Set up and solve inequalities involving derivatives to identify monotonic intervals.
- Prove that a given function is strictly increasing or decreasing across its entire domain.
Differentiation (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- Differentiating f(x) to obtain f'(x), for example f(x) = x² − 4x gives f'(x) = 2x − 4.
- Setting up the inequality f'(x) ≥ 0 for increasing or f'(x) ≤ 0 for decreasing, then solving it.
- Solving the resulting quadratic or polynomial inequality, e.g. 2x − 4 ≥ 0 gives x ≥ 2, so f is increasing for x ≥ 2.
- Stating the exact interval, using correct open or closed endpoints, and justifying with the sign of f'(x) on each side of critical points.
- Recognising that f'(x) > 0 is sufficient for strictly increasing but not necessary, using f(x) = x³ at x = 0 as a counterexample.
Examiner Tips
- 💡Remember: f'(x) ≥ 0 indicates increasing and f'(x) ≤ 0 indicates decreasing; use f'(x) > 0 or f'(x) < 0 when strict monotonicity is required.
- 💡Sketch a quick parabola of f'(x) to identify whether the inequality requires an inner or outer interval.
Common Mistakes
- Setting f(x) ≥ 0 instead of f'(x) ≥ 0 when finding where a function is increasing; correction: it is the derivative's sign, not the function's value, that determines monotonicity.
- Believing that if f'(c) = 0 at a point then f cannot be strictly increasing there; correction: f(x) = x³ is strictly increasing on [-1, 1] yet f'(0) = 0, so a zero derivative at an isolated point does not prevent strict increase.
- Forgetting to reverse inequality signs when dividing by a negative coefficient; correction: dividing by a negative number flips the inequality, e.g. −2x ≥ 0 gives x ≤ 0.