IGCSE Further Pure Mathematics key terms

99 terms with short definitions you can learn for the exam. Type to filter the list.

Key terms glossary 62

Acceleration: Rate of change of velocity, a = dv/dt.
Arithmetic series: A series where each term is the previous term plus a fixed common difference d.
Asymptote: A line that a curve gets closer and closer to but never meets, such as y = 0 for y = e^{x}.
Binomial expansion: Writing (1 + x)^{n} as a series of powers of x, using the coefficients 1, n, n(n − 1)/2! and so on.
Chain rule: The rule for differentiating a function of a function: derivative of the outside times derivative of the inside.
Change of base: Rewriting a logarithm in another base: log_{a} x = log_{b} x/log_{b} a.
Collinear: Points that lie on the same straight line. In vectors: two vectors are parallel and share a point.
Common difference: The fixed amount d added to get from one term to the next in an arithmetic series.
Common ratio: The fixed number r each term is multiplied by in a geometric series.
Completing the square: Writing ax^{2} + bx + c in the form a(x + p)^{2} + q, which shows the turning point.
Convergent series: An infinite series whose sum approaches a fixed value. A geometric series converges when |r| < 1.
Cosine rule: a^{2} = b^{2} + c^{2} − 2bc cos A, used with two sides and the included angle, or three sides.
Critical values: The values where an inequality's expression equals zero; they mark the ends of the solution intervals.
Cubic equation: A polynomial equation whose highest power is x^{3}.
Definite integral: An integral with limits, ∫_{a}^{b} f(x) dx, which gives a number (for example an area).
Derivative: The gradient function dy/dx: the rate of change of y with respect to x.
Discriminant: b^{2} − 4ac for ax^{2} + bx + c = 0. Positive: two real roots; zero: equal roots; negative: no real roots.
Displacement: Distance from a fixed point in a given direction, usually s in kinematics.
Exponential function: A function such as y = a^{x} or y = e^{x}, where the variable is in the power.
Factor theorem: If f(a) = 0 then (x − a) is a factor of f(x).
Geometric series: A series where each term is the previous term multiplied by a common ratio r.
Gradient: The steepness of a line: change in y divided by change in x.
Identity: A statement true for all values of the variable, often written with ≡.
Included angle: The angle between two given sides of a triangle.
Indefinite integral: An integral without limits; the answer includes a constant of integration, + c.
Kinematics: The study of motion using displacement s, velocity v and acceleration a, linked by calculus.
Linear programming: Finding the greatest or least value of an expression subject to linear inequalities, by checking the vertices of the region.
Logarithm: log_{a} x is the power a must be raised to to give x: log_{a} x = y means a^{y} = x.
Magnitude: The length of a vector: |ai + bj| = √{a^{2} + b^{2}}.
Maximum point: A stationary point where the curve changes from increasing to decreasing; d^{2}y/dx^{2} < 0.
Minimum point: A stationary point where the curve changes from decreasing to increasing; d^{2}y/dx^{2} > 0.
Natural logarithm: ln x, the logarithm to base e.
Normal: The line at a point on a curve that is perpendicular to the tangent there; gradient −1/m.
Perpendicular: At right angles. Perpendicular lines have gradients whose product is −1.
Polynomial: An expression made of whole-number powers of x, such as 2x^{3} − x + 5.
Position vector: The vector from the origin O to a point, written a for point A.
Principal value: The first solution a calculator gives for an inverse trig function.
Product rule: d/dx(uv) = udv/dx + vdu/dx.
Quotient rule: d/dx(u/v) = [(vdu/dx − udv/dx)/v^{2}].
Radian: The angle at the centre when the arc length equals the radius. π radians = 180°.
Rate of change: How fast one quantity changes compared with another, found by differentiating; linked using the chain rule.
Rational function: A function written as one polynomial divided by another, such as y = (ax + b)/(cx + d).
Rationalise the denominator: Remove a surd from the bottom of a fraction by multiplying top and bottom by a suitable surd expression.
Remainder theorem: The remainder when f(x) is divided by (x − a) is f(a).
Resultant: The single vector equal to the sum of two or more vectors.
Roots: The solutions of an equation f(x) = 0; where the graph crosses the x-axis.
Scalar: A quantity with size only, such as speed or distance.
Second derivative: d^{2}y/dx^{2}, the derivative of dy/dx; used to decide the nature of stationary points.
Sector: The part of a circle between two radii and an arc; area 1/2r^{2}θ with θ in radians.
Sigma notation: Using Σ to write a sum, for example Σ_{r = 1}^{n} r means 1 + 2 + ... + n.
Sine rule: a/sin A = b/sin B = c/sin C, used with a side and its opposite angle.
Small change: An approximation δy ≈ dy/dx δx for a small change δx.
Stationary point: A point where dy/dx = 0: a maximum, a minimum or a point of inflexion.
Sum and product of roots: For ax^{2} + bx + c = 0: α + β = −b/a and αβ = c/a.
Sum to infinity: The limit of a convergent geometric series: S_{∞} = a/(1 − r), valid when |r| < 1.
Surd: An irrational root left in exact form, such as √{2}.
Tangent: A straight line that touches a curve at a point, with the same gradient as the curve there.
Unit vector: A vector of length 1, found by dividing a vector by its magnitude.
Validity: The range of x for which an infinite binomial series converges: (1 + kx)^{n} needs |kx| < 1.
Vector: A quantity with size and direction, such as displacement or velocity.
Velocity: Rate of change of displacement, v = ds/dt.
Volume of revolution: The volume made when a region is rotated 360° about an axis: π∫y^{2} dx about the x-axis.

Formulae given on the exam sheet 14

Addition formulae for cos: cos(A ± B) = cos A cos B ∓ sin A sin B
Addition formulae for sin: sin(A ± B) = sin A cos B ± cos A sin B
Addition formulae for tan: tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Binomial series: (1 + x)^{n} = 1 + nx + n(n − 1)/2!x^{2} + ... + n(n − 1)...(n − r + 1)/r!x^{r} + ..., for |x| < 1, n any real number
Change of base of logarithms: log_{a} x = log_{b} x/log_{b} a
Cosine rule: a^{2} = b^{2} + c^{2} − 2bc cos A
Curved surface area of a cone: πr × slant height
Quotient rule: d/dx(f(x)/g(x)) = [(f'(x)g(x) − f(x)g'(x))/[g(x)]^{2}]
Sum of a finite geometric series: S_{n} = a(1 − r^{n})/(1 − r)
Sum of an arithmetic series: S_{n} = n/2[2a + (n − 1)d]
Sum to infinity of a geometric series: S_{∞} = a/(1 − r), for |r| < 1
Surface area of a sphere: 4πr^{2}
tan θ: tan θ = sin θ/cos θ
Volume of a sphere: 4/3πr^{3}

Formulae you must learn 23

Arc length and sector area: s = rθ, A = 1/2r^{2}θ (θ in radians)
Area between two curves: ∫_{a}^{b} [g(x) − f(x)] dx, where g(x) ≥ f(x)
Area of a triangle: 1/2ab sin C
Area under a curve: ∫_{a}^{b} y dx (for y ≥ 0)
Chain rule: d/dx[f(g(x))] = f'(g(x))g'(x)
Differentiating sin ax, cos ax, e^{ax}: sin ax → a cos ax; cos ax → −a sin ax; e^{ax} → ae^{ax} (x in radians)
Differentiating x^{n}: d/dx(x^{n}) = nx^{n − 1}
Distance between two points: d = √{(x_{2} − x_{1})^{2} + (y_{2} − y_{1})^{2}}
Equation from its roots: x^{2} − (α + β)x + αβ = 0
Gradient of a line: m = (y_{2} − y_{1})/(x_{2} − x_{1})
Integrating sin ax, cos ax, e^{ax}: ∫sin ax dx = −1/acos ax + c; ∫cos ax dx = 1/asin ax + c; ∫e^{ax} dx = 1/ae^{ax} + c
Integrating x^{n}: ∫x^{n} dx = x^{n + 1}/(n + 1) + c, n ≠ −1
Laws of logarithms: log_{a} xy = log_{a} x + log_{a} y; log_{a} x/y = log_{a} x − log_{a} y; log_{a} x^{k} = k log_{a} x
nth term of a geometric series: ar^{n − 1}
nth term of an arithmetic series: l = a + (n − 1)d
Point dividing a line in the ratio m : n: ((nx_{1} + mx_{2})/(m + n), (ny_{1} + my_{2})/(m + n))
Product rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Pythagorean identity: cos^{2} θ + sin^{2} θ = 1
Quadratic formula: x = (−b ± √{b^{2} − 4ac})/2a for ax^{2} + bx + c = 0
Sine rule: a/sin A = b/sin B = c/sin C
Special logarithms: log_{a} a = 1, log_{a} 1 = 0, log_{a} x = 1/log_{x} a
Sum and product of roots: α + β = −b/a, αβ = c/a
Volume of revolution: π∫_{a}^{b} y^{2} dx about the x-axis; π∫_{c}^{d} x^{2} dy about the y-axis

Practise Further Pure Mathematics free Revision checklist