A-Level Mathematics Flashcards

What is the derivative of f(x) = x⁴ − 3x² + 5x − 2?

Answer: 4x³ − 6x + 5

Using the power rule: d/dx(xⁿ) = nxⁿ⁻¹. d/dx(x⁴) = 4x³, d/dx(−3x²) = −6x, d/dx(5x) = 5, d/dx(−2) = 0. So f'(x) = 4x³ − 6x + 5.

What does ∫(3x² + 2x) dx equal?

Answer: x³ + x² + C

Integrate term by term: ∫3x² dx = x³ and ∫2x dx = x². Add constant C. So ∫(3x² + 2x) dx = x³ + x² + C.

What is the binomial expansion of (1 + x)ⁿ for small x?

Answer: 1 + nx + n(n−1)/2! · x² + n(n−1)(n−2)/3! · x³ + ...

The binomial expansion is (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ... This is valid for |x| < 1 when n is not a positive integer. For positive integers it terminates.

Solve: 2sin²θ − sinθ − 1 = 0 for 0° ≤ θ ≤ 360°

Answer: θ = 90°, 210°, 330°

Let u = sinθ: 2u² − u − 1 = 0 → (2u + 1)(u − 1) = 0. So sinθ = −1/2 or sinθ = 1. sinθ = 1 → θ = 90°. sinθ = −1/2 → θ = 210°, 330°. So θ = 90°, 210°, 330°.

What is the chain rule for differentiation?

Answer: dy/dx = dy/du × du/dx

The chain rule states: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x), or equivalently dy/dx = (dy/du)(du/dx). It is used to differentiate composite functions.

What is the sum of an infinite geometric series with first term a and common ratio r (|r| < 1)?

Answer: a/(1 − r)

For an infinite geometric series where |r| < 1, the sum S∞ = a/(1 − r). This converges because successive terms get smaller and smaller, approaching zero.