Topic 3 of 8 · Form 5
Pure 1: series, differentiation and integration
Binomial expansion of (a + b)ⁿ for positive integer n; arithmetic and geometric progressions; sum to infinity |r| < 1.
Notes for this topic 1 file, in order
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Series
Binomial expansion of (a + b)ⁿ for positive integer n; arithmetic and geometric progressions; sum to infinity |r| < 1.
Worked example: Find the coefficient of x² in (1 − 2x)⁵(2 + x).
(1 − 2x)⁵ = 1 − 10x + 40x² + …; x² terms: 2 × 40x² + x × (−10x) = 70x² → coefficient 70
Differentiation
Power rule, chain rule; gradients, tangents and normals; increasing and decreasing functions; stationary points and their nature; connected rates of change.
Integration
Reverse of differentiation for powers and (ax + b)ⁿ; finding constants; definite integrals; area under curves and between curves; volumes of revolution about the x- and y-axes (V = π∫y² dx).
Worked example: Find the volume generated when the region under y = √(x + 1) from x = 0 to x = 3 is rotated 360° about the x-axis.
V = π∫₀³(x + 1) dx = π[x²/2 + x]₀³ = π(4.5 + 3) = 7.5π ≈ 23.6
Check yourself
- Find the sum to infinity of 18 + 12 + 8 + …
- Differentiate (3x² + 1)⁴.
- Find ∫ 6/(2x + 1)³ dx.
Show answers
- 18 ÷ (1 − ⅔) = 54
- 24x(3x² + 1)³
- −3/(2(2x + 1)²) + c
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