SPM Add Maths flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Composite function fg

Defined by (fg)(x) = f(g(x)); apply g first, then f. It exists only when the range of g lies within the domain of f.

Inverse function f⁻¹

Exists only for one-to-one functions. Swap x and y in y = f(x) and solve for y; its graph mirrors f in the line y = x.

Discriminant b² − 4ac

Determines the nature of roots of ax² + bx + c = 0: positive gives two distinct real roots, zero gives equal roots, negative gives no real roots.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, α + β = −b/a and αβ = c/a. Used to build new quadratics from known roots.

Vertex form of quadratic

y = a(x − h)² + k gives vertex (h, k) and axis of symmetry x = h. Positive a gives a minimum, negative a a maximum.

Linear–quadratic simultaneous equations

Solve the linear equation for one variable, substitute into the quadratic, then solve. Discriminant of the resulting quadratic gives the number of intersection points.

No-solution linear system

Two linear equations in two unknowns have no solution when their graphs are parallel: equal gradients but different y-intercepts, so they never intersect.

Logarithm definition

log_a N = x means aˣ = N, valid for a > 0, a ≠ 1 and N > 0. It answers the question: to what power must a be raised?

Change of base formula

log_a b = log_c b / log_c a for any valid base c. A common special case is log_a b = 1 / log_b a.

Solving aˣ = b

Take logarithms of both sides to obtain x = log b / log a. Any base may be used, typically base 10 or e.

Perpendicular gradient condition

Two non-vertical lines are perpendicular exactly when the product of their gradients is −1, so m₂ = −1/m₁.

Midpoint formula

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2), the average of the coordinates.

Arc length formula

For a circle of radius r and angle θ measured in radians, arc length s = rθ. Degrees must be converted before use.

Sector area formula

Area of a sector is A = ½r²θ with θ in radians, equivalently A = ½rs where s is the arc length.

Area of a circular segment

Segment area equals sector area minus triangle area: ½r²θ − ½r² sin θ, with θ in radians at the centre.

Chain rule

d/dx[f(g(x))] = f′(g(x)) × g′(x). Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Quotient rule

For y = u/v, dy/dx = (v du/dx − u dv/dx)/v². The numerator order matters; reversing it changes the sign of the answer.

Second derivative test

At a stationary point where f′(x) = 0: f″(x) > 0 indicates a minimum, f″(x) < 0 indicates a maximum, f″(x) = 0 is inconclusive.

Definite integral evaluation

∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f. The constant of integration cancels, so it is omitted.

Volume of revolution

Rotating y = f(x) about the x-axis between x = a and x = b gives volume V = π∫ₐᵇ y² dx.

Area between two curves

Integrate upper curve minus lower curve between the x-values of intersection: ∫ₐᵇ (y_upper − y_lower) dx.

Unit vector

The unit vector in the direction of a is â = a/|a|. It has magnitude 1 and preserves the direction of the original vector.

Dot product of vectors

a·b = |a||b| cos θ = x₁x₂ + y₁y₂. A zero dot product means the two vectors are perpendicular.

Position vector subtraction

For points A and B with position vectors OA and OB, the vector AB = OB − OA, pointing from A to B.

Cosine rule

a² = b² + c² − 2bc cos A. Use it with three sides, or two sides and the included angle, to find the remaining side or angle.

Sine rule

a/sin A = b/sin B = c/sin C. Use it with a matching side–angle pair plus one more known value, watching for the ambiguous case.

Ambiguous case of sine rule

Given two sides and a non-included angle, the triangle may have zero, one, or two valid solutions because the sine of an angle is positive for two supplementary angles.

Index number formula

I = (Q₁/Q₀) × 100, where Q₀ is the base-year value and Q₁ the current value. The base year always has index 100.

Composite index

A weighted average of component indices: I = Σ(W × I)/ΣW, where W is each weight and I the corresponding index number.

Velocity and acceleration

Velocity is the rate of change of displacement, v = ds/dt. Acceleration is the rate of change of velocity, a = dv/dt = d²s/dt².

Turning point of a moving particle

A particle reverses direction where its velocity is zero. Total distance requires summing the absolute displacements of each interval between turning points.

Maximum velocity condition

Velocity reaches a maximum or minimum where acceleration is zero, a = dv/dt = 0, provided the acceleration changes sign there.