A-Level Mathematics Flashcards

What is integration by parts used for?

Answer: Integrating a product of two functions: ∫uv' dx = uv − ∫u'v dx

Integration by parts is used when integrating a product of two functions. The formula is ∫u(dv/dx)dx = uv − ∫v(du/dx)dx. Choose u as the function that simplifies when differentiated (use LIATE: Log, Inverse trig, Algebra, Trig, Exponential).

What is the equation of a tangent to the curve y = x² at the point (2, 4)?

Answer: y = 4x − 4

dy/dx = 2x. At x = 2, gradient = 4. Tangent equation: y − 4 = 4(x − 2) → y = 4x − 4.

What is Euler's formula: e^(iθ) = ?

Answer: cos θ + i sin θ

Euler's formula states e^(iθ) = cos θ + i sin θ. At θ = π, this gives e^(iπ) + 1 = 0 (Euler's identity). It links complex numbers, exponentials, and trigonometry in one elegant equation.

What is the formula for the sum of the first n terms of an arithmetic series?

Answer: Sₙ = n(a + l)/2

Sum of arithmetic series: Sₙ = n/2(a + l) = n/2(2a + (n−1)d), where a is the first term, l is the last term, d is the common difference, and n is the number of terms.

What is the condition for two lines in 3D to be perpendicular?

Answer: The dot product of their direction vectors equals zero

Two lines with direction vectors a and b are perpendicular if a · b = 0 (dot product equals zero). The dot product a · b = |a||b|cos θ; if θ = 90°, cos 90° = 0, so a · b = 0.

What does the trapezium rule estimate?

Answer: The area under a curve by dividing it into trapezoids

The trapezium rule approximates the area under a curve by dividing the region into trapezoids. Area ≈ h/2[y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ], where h is the strip width. More strips gives greater accuracy.