Numerical Methods

The equation x³ − 2x − 5 = 0 has a root in which of the following intervals, found by a change of sign?

• Abetween x = 0 and x = 1
• Bbetween x = 1 and x = 2
• Cbetween x = 2 and x = 3✓
• Dbetween x = 3 and x = 4

Answer: Let f(x) = x³ − 2x − 5. Then f(2) = 8 − 4 − 5 = −1 (negative) and f(3) = 27 − 6 − 5 = 16 (positive). The sign change shows a root lies between x = 2 and x = 3.

The iteration xₙ₊₁ = √(2xₙ + 3) is used with x₀ = 2. Find x₁ correct to 3 decimal places.

• A2.000
• B3.500
• C1.871
• D2.646✓

Answer: x₁ = √(2 × 2 + 3) = √7 = 2.6457…, which is 2.646 to 3 decimal places.

The Newton-Raphson method is applied to f(x) = x² − 5 with x₀ = 2. Find x₁.

• A2.25✓
• B2.5
• C1.75
• D2.2

Answer: f'(x) = 2x. Then x₁ = x₀ − f(x₀)/f'(x₀) = 2 − (2² − 5)/(2 × 2) = 2 − (−1)/4 = 2 + 0.25 = 2.25.

Which formula correctly states the Newton-Raphson iteration for solving f(x) = 0?

• Axₙ₊₁ = xₙ + f(xₙ)/f'(xₙ)
• Bxₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)✓
• Cxₙ₊₁ = xₙ − f'(xₙ)/f(xₙ)
• Dxₙ₊₁ = f(xₙ)/f'(xₙ)

Answer: The Newton-Raphson formula is xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ), subtracting the ratio of the function to its derivative at the current estimate.

Use the trapezium rule with 2 strips of width 1 to estimate the area under y = x² from x = 0 to x = 2. The ordinates at x = 0, 1, 2 are 0, 1, 4.

• A5
• B4
• C3✓
• D2.5

Answer: Trapezium rule: area ≈ (h/2)[y₀ + 2y₁ + y₂] = (1/2)[0 + 2(1) + 4] = (1/2)(6) = 3.

The trapezium rule is used to estimate the area under the curve y = x², which is convex (it bends upward). The estimate will be:

• Aan underestimate, because the tops of the strips lie below the curve
• Bexactly the true area, because the curve is a polynomial
• Can underestimate, because the curve is increasing
• Dan overestimate, because the chords lie above the convex curve✓

Answer: For a curve that is convex (concave up), each chord joining the tops of a strip lies above the curve, so the trapezia include extra area. The trapezium rule therefore gives an overestimate.