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Topic Key:
Functions
Quadratics
Polynomials
Binomial
Log / Exp
Trigonometry
Differentiation
Integration
A
Absolute Value
Functions
Another name for the modulus. |x| = x if x ≥ 0 and |x| = −x if x < 0. Always non-negative. The graph of y = |f(x)| reflects any portion below the x-axis upward above it.
Addition Formulae
Trigonometry
Formulae expressing sin(A±B), cos(A±B), tan(A±B) in terms of sinA, cosA, sinB, cosB. Key: sin(A+B)=sinA cosB+cosA sinB, cos(A+B)=cosA cosB−sinA sinB. Used to find exact values and prove identities.
Antiderivative
Integration
A function F(x) whose derivative is f(x). Also called a primitive or indefinite integral. The general antiderivative includes an arbitrary constant c: ∫f(x)dx = F(x)+c. Integration is the process of finding antiderivatives.
Arc Length (calculus)
Integration
Not directly in 4037/0606 syllabus, but the concept of area under a curve is central. Area between curve y=f(x) and x-axis from a to b = ∫ₐᵇ f(x)dx. Negative when curve is below the x-axis — take modulus for actual area.
Asymptote
Log / Exp
A line that a curve approaches but never reaches. For y=eˣ: horizontal asymptote y=0. For y=ln x: vertical asymptote x=0. For y=eˣ+k: asymptote y=k. For y=ln(x+a): asymptote x=−a. Asymptotes must be stated when sketching these graphs.
B
Binomial Coefficient
Binomial
The value ⁿCᵣ = n!/[r!(n−r)!] — the number of ways to choose r items from n. Appears as the coefficient in the binomial expansion. Symmetry: ⁿCᵣ = ⁿCₙ₋ᵣ. Special values: ⁿC₀ = ⁿCₙ = 1.
Binomial Theorem
Binomial
The formula for expanding (a+b)ⁿ for any positive integer n: (a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ for r=0 to n. Produces (n+1) terms. The general (r+1)th term is T_{r+1} = ⁿCᵣ aⁿ⁻ʳ bʳ.
C
Chain Rule
Differentiation
The rule for differentiating composite functions: dy/dx = (dy/du)(du/dx). For y=[f(x)]ⁿ: dy/dx = n[f(x)]ⁿ⁻¹ × f'(x). Applied whenever one function is nested inside another — differentiate the outside keeping the inside unchanged, then multiply by the derivative of the inside.
Change of Base
Log / Exp
Formula to convert between logarithm bases: logₐ x = log x / log a = ln x / ln a. Allows evaluation of any logarithm using a calculator (which has log₁₀ and ln). Also used to compare or combine logs with different bases.
Codomain
Functions
The set of all possible output values of a function — also called the range. For f: X→Y, the codomain is Y. The range is the actual set of values produced by f — the range is a subset of the codomain.
Completing the Square
Quadratics
Rewriting ax²+bx+c in the form a(x+p)²+q. Method: factor a from first two terms, add and subtract (b/2a)² inside. Used to find the vertex/turning point (−p, q), prove always positive/negative, and solve quadratic equations. For a(x+p)²+q: minimum value q occurs at x=−p (if a>0).
Composite Function
Functions
fg(x) = f(g(x)) — g is applied FIRST, then f is applied to the result. The domain of fg is the set of x in domain(g) such that g(x) lies in domain(f). fg ≠ gf in general. f²(x) = ff(x) means f applied twice.
Concavity
Differentiation
The direction a curve bends. If d²y/dx² > 0 at a point, the curve is concave upward (∪ shape) — the gradient is increasing. If d²y/dx² < 0, the curve is concave downward (∩ shape). Changes in concavity occur at points of inflection.
Connected Rates of Change
Differentiation
Using the chain rule to relate rates of change of different quantities with respect to time: dy/dt = (dy/dx)(dx/dt). Used when the rate of change of one variable (e.g. area, volume) is linked to another (e.g. radius, depth) through a geometric relationship.
Constant of Integration
Integration
The arbitrary constant c added to every indefinite integral. Since differentiating any constant gives zero, infinitely many antiderivatives exist — they all differ by a constant. c is determined when additional information is given (e.g. a point on the curve or initial condition). Never omit c in indefinite integration.
Cosecant (cosec)
Trigonometry
The reciprocal of sine: cosec θ = 1/sin θ. Undefined when sin θ = 0 (i.e. θ = 0°, 180°, 360°, ...). Appears in the identity 1+cot²θ = cosec²θ (obtained by dividing sin²θ+cos²θ=1 by sin²θ). Used in identity proofs.
Cotangent (cot)
Trigonometry
The reciprocal of tangent: cot θ = cos θ / sin θ = 1/tan θ. Undefined when sin θ = 0. Appears in the identity 1+cot²θ=cosec²θ. Less common than sec and cosec but appears in identity proofs — always convert to sin/cos if stuck.
Critical Point
Differentiation
A point where dy/dx = 0 or dy/dx is undefined. Stationary points (dy/dx=0) include local maxima, local minima, and points of inflection. Used in optimisation to find maximum and minimum values of a function in real-world problems.
D
Decreasing Function
Differentiation
A function f is decreasing on an interval if dy/dx < 0 for all x in that interval — the function values fall as x increases. Between a local maximum and the next local minimum, a continuous function is decreasing.
Definite Integral
Integration
An integral with specific limits: ∫ₐᵇ f(x)dx = F(b)−F(a) where F is any antiderivative of f. Gives a numerical value. Represents the signed area between y=f(x) and the x-axis from x=a to x=b. No constant of integration needed.
Degree (polynomial)
Polynomials
The highest power of x in a polynomial. A polynomial of degree n has at most n real roots and at most (n−1) turning points. For partial fractions: if degree of numerator ≥ degree of denominator, the fraction is improper and must be divided first.
Derivative
Differentiation
The instantaneous rate of change of a function — the gradient of the tangent at any point. Written dy/dx, f'(x), or d/dx[f(x)]. The second derivative d²y/dx² is the derivative of dy/dx — measures how the gradient itself changes.
Discriminant
Quadratics
The expression Δ = b²−4ac in the quadratic ax²+bx+c. Δ > 0: two distinct real roots. Δ = 0: one repeated root (tangent to x-axis). Δ < 0: no real roots. Used to prove always positive/negative and to find conditions on parameters k for different root types.
Domain
Functions
The complete set of permitted input values (x-values) for a function. Must be stated explicitly. Key restrictions: √g(x) requires g(x)≥0; 1/g(x) requires g(x)≠0; ln(g(x)) requires g(x)>0. Domain of f⁻¹ equals Range of f.
Double Angle Formulae
Trigonometry
Special cases of addition formulae with A=B: sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A). cos2A has three forms — choose the form that matches the equation to be solved.
E
e (Euler's Number)
Log / Exp
The irrational constant e ≈ 2.71828... — the base of natural logarithms. The function y=eˣ is uniquely self-differentiating: d/dx(eˣ)=eˣ. This makes it the most mathematically natural exponential function. Also: e = lim[n→∞](1+1/n)ⁿ.
Exponential Function
Log / Exp
A function of the form y=aˣ where a>0, a≠1. For a>1: strictly increasing, models growth. For 0<a<1: strictly decreasing, models decay. All pass through (0,1). Range: y>0. The most important is y=eˣ. Never crosses the x-axis.
Extraneous Solution
Log / Exp
A solution found algebraically that does not satisfy the original equation. Common in logarithmic equations — a value that makes a log argument zero or negative must be rejected. Always substitute solutions back and check all log arguments are strictly positive.
F
Factor Theorem
Polynomials
A special case of the Remainder Theorem: (x−a) is a factor of polynomial f(x) if and only if f(a)=0. For (ax−b): substitute x=b/a. Used to find factors of polynomials, verify factors, and determine unknown coefficients when a factor is given.
First Derivative Test
Differentiation
A method to classify stationary points using the sign of dy/dx on either side. If dy/dx changes +→−: local maximum. If dy/dx changes −→+: local minimum. If dy/dx does not change sign: point of inflection. Used when d²y/dx²=0 (inconclusive second derivative test).
First Principles
Differentiation
Deriving the derivative from the definition: f'(x) = lim[h→0][f(x+h)−f(x)]/h. The gradient of the chord from (x,f(x)) to (x+h, f(x+h)) as h→0 becomes the gradient of the tangent. Cambridge 4037/0606 may require differentiation of simple functions from first principles.
Function
Functions
A rule that maps each element of the domain to exactly one element of the range. Written f(x) or f:x↦f(x). A relation is only a function if every input gives exactly one output. Many-to-one functions are valid (e.g. f(x)=x²) but one-to-many is NOT a function.
G
General Solution
Trigonometry
The complete set of all solutions to a trig equation expressed using integer n: sin θ=k → θ=nπ+(−1)ⁿα; cos θ=k → θ=2nπ±α; tan θ=k → θ=nπ+α, where α is the principal value and n∈ℤ. In degrees: replace π with 180°.
General Term
Binomial
The formula for the (r+1)th term in a binomial expansion: T_{r+1} = ⁿCᵣ aⁿ⁻ʳ bʳ where r starts from 0. Used to find any specific term, the coefficient of xᵏ (set power of x equal to k and solve for r), or the constant term (set total power of x to 0).
Gradient Function
Differentiation
The derivative dy/dx — a function that gives the gradient of the curve at any point x. The gradient of the tangent at (a, f(a)) is dy/dx evaluated at x=a. Used to find equations of tangents, normals, stationary points, and intervals of increase/decrease.
I
Identity (Trigonometric)
Trigonometry
An equation true for all valid values of the variable, written with ≡. Key identities: sin²θ+cos²θ≡1, 1+tan²θ≡sec²θ, 1+cot²θ≡cosec²θ. To prove an identity, work on ONE side only — transform it to match the other side without crossing the ≡ sign.
Improper Fraction
Polynomials
A rational function where the degree of the numerator ≥ degree of the denominator. Must be divided first (polynomial long division) before decomposing into partial fractions. The result is a polynomial plus a proper fraction, which is then decomposed normally.
Increasing Function
Differentiation
A function f is increasing on an interval if dy/dx > 0 for all x in that interval — function values rise as x increases. Between a local minimum and the next local maximum, a continuous function is increasing. Found by solving dy/dx > 0.
Indefinite Integral
Integration
An integral without limits: ∫f(x)dx = F(x)+c. Represents a family of functions all differing by the constant c. The constant c is determined from additional information (e.g. curve passes through a given point). Always include +c.
Integration by Substitution
Integration
A technique for integrating composite functions. Choose u=g(x), find du/dx, replace dx=du/(du/dx), substitute — all x's must disappear. Integrate with respect to u, then substitute back. For definite integrals, change the limits using u=g(a) and u=g(b).
Inverse Function
Functions
f⁻¹(x) reverses the effect of f. Exists only if f is one-to-one. Method: write y=f(x), swap x↔y, solve for y. Key properties: ff⁻¹(x)=x and f⁻¹f(x)=x. Domain of f⁻¹=Range of f. The graph of f⁻¹ is the reflection of f in the line y=x.
K
Kinematics
Differentiation
The study of motion in a straight line using calculus. Displacement s, velocity v=ds/dt, acceleration a=dv/dt=d²s/dt². v>0: moving in positive direction; v<0: negative direction; v=0: at rest. Distance ≠ displacement — split at v=0 and sum absolute values of each section.
L
Laws of Logarithms
Log / Exp
Four rules: Product: logₐ(xy)=logₐx+logₐy; Quotient: logₐ(x/y)=logₐx−logₐy; Power: logₐ(xⁿ)=n logₐx; Change of base: logₐx=logx/loga. Common error: log(x+y)≠logx+logy — only products split into sums.
Linearisation
Log / Exp
Converting a non-linear relationship to the form Y=mX+c by taking logarithms. y=abˣ → lgy=x lgb+lga (plot lgy vs x); y=axⁿ → lgy=n lgx+lga (plot lgy vs lgx); y=aeᵇˣ → lny=bx+lna (plot lny vs x). Gradient and intercept give the unknown constants.
Local Maximum
Differentiation
A stationary point where the function value is greater than at nearby points. At a local max: dy/dx=0, d²y/dx²<0 (curve concave down). The gradient changes from positive to negative through the point. Not necessarily the overall (global) maximum of the function.
Local Minimum
Differentiation
A stationary point where the function value is less than at nearby points. At a local min: dy/dx=0, d²y/dx²>0 (curve concave up). The gradient changes from negative to positive through the point. Not necessarily the overall (global) minimum.
Logarithm
Log / Exp
The inverse of an exponential: logₐx=y ⟺ aʸ=x. The common logarithm (log or log₁₀) has base 10. The natural logarithm (ln or logₑ) has base e. Key values: logₐ1=0 (since a⁰=1); logₐa=1 (since a¹=a); logₐaˣ=x; a^(logₐx)=x.
M
Many-to-One Function
Functions
A function where two or more different inputs give the same output. Examples: f(x)=x², f(x)=cosx, f(x)=|x|. Many-to-one functions do NOT have an inverse over their full domain. To create an inverse, restrict the domain so the function becomes one-to-one.
Modulus Function
Functions
|x| = x if x≥0; −x if x<0. Always non-negative. Solving |f(x)|=k: two cases f(x)=k or f(x)=−k. Solving |f(x)|<k: −k<f(x)<k (single interval). Solving |f(x)|>k: f(x)>k or f(x)<−k (two separate regions). Always check solutions.
N
Natural Logarithm
Log / Exp
The logarithm base e: ln x = logₑ x. The inverse of y=eˣ. Domain: x>0. Range: all reals. Graph passes through (1,0) with vertical asymptote x=0. d/dx(ln x)=1/x. ∫(1/x)dx=ln|x|+c. Used in solving exponential equations and in integration.
Normal (to a curve)
Differentiation
The straight line perpendicular to the tangent at a given point on a curve. If the tangent has gradient m, the normal has gradient −1/m (negative reciprocal). Equation: y−y₁=(−1/m)(x−x₁). If the tangent is horizontal (m=0), the normal is vertical: x=x₁.
O
One-to-One Function
Functions
A function where each output value corresponds to exactly one input — no two different inputs give the same output. Test: any horizontal line crosses the graph at most once (horizontal line test). One-to-one functions have an inverse function. Examples: f(x)=2x+3, f(x)=eˣ, f(x)=x³.
Optimisation
Differentiation
Finding maximum or minimum values in real-world problems. Method: define variables, write constraint and objective function, substitute to get one-variable function, differentiate and set to zero, solve, verify using d²y/dx² or sign test, answer in context with units.
P
Partial Fractions
Polynomials
Decomposing a rational function into a sum of simpler fractions. Three types: distinct linear factors (A/(ax+b)+B/(cx+d)); repeated factor (A/(ax+b)+B/(ax+b)²+C/(cx+d)); irreducible quadratic ((Ax+B)/(ax²+bx+c)+C/(dx+e)). Essential for integration of rational functions.
Pascal's Triangle
Binomial
A triangular array of numbers where each entry is the sum of the two above it. Row n gives the binomial coefficients for (a+b)ⁿ. Efficient for small n, but the nCr formula is needed for large n. The symmetry of Pascal's Triangle reflects the identity ⁿCᵣ=ⁿCₙ₋ᵣ.
Point of Inflection
Differentiation
A point where the concavity of a curve changes — d²y/dx² changes sign. At a stationary point of inflection: dy/dx=0 AND d²y/dx²=0 AND the sign of d²y/dx² changes. A zero second derivative alone does NOT confirm a point of inflection — always check for a sign change.
Polynomial
Polynomials
An expression of the form aₙxⁿ+aₙ₋₁xⁿ⁻¹+...+a₁x+a₀ where n is a non-negative integer. The degree is the highest power. Polynomials of degree 1 are linear, degree 2 are quadratic, degree 3 are cubic. Operations: addition, subtraction, multiplication, and long division are all defined.
Product Rule
Differentiation
For y=uv: dy/dx = u(dv/dx)+v(du/dx). Used when two functions are multiplied and neither can be simplified. Common error: multiplying derivatives (u'v' is WRONG). After differentiating, factorise the result before finding stationary points.
Q
Quadratic Inequality
Quadratics
An inequality of the form ax²+bx+c>0, <0, ≥0, or ≤0. Method: factorise → identify roots p and q (p<q) → sketch parabola → identify regions. For a>0: f(x)<0 between roots (p<x<q); f(x)>0 outside roots (x<p or x>q). Always sketch — never just solve algebraically.
Quotient Rule
Differentiation
For y=u/v: dy/dx=(v·du/dx − u·dv/dx)/v². Memory: "bottom×diff(top) minus top×diff(bottom), over bottom squared." The order in the numerator matters — getting it reversed gives the wrong sign. Verify by checking the formula: v(u')−u(v') not u(v')−v(u').
R
R Form
Trigonometry
Writing a sinx+b cosx as R sin(x+α) where R=√(a²+b²) and tanα=b/a (α acute). Method: expand R sin(x+α) by addition formula, compare coefficients. Maximum=R, minimum=−R. Used to find max/min values, solve equations, and minimise expressions like 1/[f(x)+k].
Range
Functions
The set of all output values (y-values) produced by a function for inputs in its domain. Found by considering what f(x) can and cannot equal. Range of f⁻¹ = Domain of f. For quadratics, use completing the square to identify the range. For eˣ: range is y>0; for ln x: range is all reals.
Remainder Theorem
Polynomials
When polynomial f(x) is divided by (x−a), the remainder equals f(a). For divisor (ax−b), the remainder equals f(b/a). This allows finding the remainder without performing long division, and is used with the Factor Theorem to find unknown coefficients by forming simultaneous equations.
Restricted Domain
Functions
A reduced subset of the natural domain, chosen to make a many-to-one function one-to-one so that its inverse exists. For f(x)=(x−a)²+b: restrict to x≥a (or x≤a) — either half of the parabola is one-to-one. The vertex x-value is always a natural boundary for restricting quadratic functions.
S
Second Derivative
Differentiation
d²y/dx² — the derivative of dy/dx. Used to classify stationary points: d²y/dx²>0 → minimum (concave up); d²y/dx²<0 → maximum (concave down); d²y/dx²=0 → inconclusive (use first derivative test). Also used in kinematics: a=d²s/dt².
Secant (sec)
Trigonometry
The reciprocal of cosine: sec θ = 1/cos θ. Undefined when cos θ=0 (i.e. θ=90°, 270°, ...). Appears in the key identity 1+tan²θ=sec²θ (from sin²θ+cos²θ=1 divided by cos²θ). Also: d/dx(tan x)=sec²x.
Self-Inverse Function
Functions
A function where f⁻¹(x)=f(x) — applying f twice returns to the original value: ff(x)=x. The graph is symmetric about the line y=x. Examples: f(x)=1/x, f(x)=−x, f(x)=(a−x)/(x−b) for certain values. To verify: show ff(x)=x OR show f⁻¹(x)=f(x) algebraically.
Stationary Point
Differentiation
A point on a curve where dy/dx=0 — the tangent is horizontal. Three types: local maximum (d²y/dx²<0), local minimum (d²y/dx²>0), stationary point of inflection (d²y/dx²=0, sign of d²y/dx² unchanged). Finding: differentiate, set to zero, solve. Classifying: use second derivative or sign test.
Substitution (integration)
Integration
See Integration by Substitution. Two key patterns to recognise: ∫f'(x)/f(x)dx = ln|f(x)|+c (f'/f form); ∫f'(x)[f(x)]ⁿdx = [f(x)]ⁿ⁺¹/(n+1)+c. These arise directly from the chain rule and are the most common substitution patterns in Cambridge 4037/0606 papers.
T
Tangent (to a curve)
Differentiation
The straight line touching a curve at a single point with gradient equal to dy/dx at that point. Equation: y−y₁=m(x−x₁) where m=dy/dx at x=x₁. The tangent and curve share the same gradient at the point of contact but generally diverge away from it.
Transformation of Graphs
Functions
Systematic changes to the graph of y=f(x): y=f(x)+a (translate up a); y=f(x+a) (translate left a); y=af(x) (vertical stretch ×a); y=f(ax) (horizontal stretch ×1/a); y=−f(x) (reflect x-axis); y=f(−x) (reflect y-axis); y=|f(x)| (reflect below-axis portions up); y=f(|x|) (keep x≥0, reflect in y-axis).
Turning Point
Differentiation
A stationary point where the gradient changes sign — either a local maximum or local minimum. NOT the same as a point of inflection (where dy/dx=0 but gradient does not change sign). Every local max/min is a turning point, but not every stationary point is a turning point.
U
Upper and Lower Bounds (integration)
Integration
The limits a and b in a definite integral ∫ₐᵇf(x)dx. The lower bound a is the left limit; the upper bound b is the right limit. Swapping limits changes the sign: ∫ₐᵇf(x)dx = −∫_b^a f(x)dx. When using substitution, change these limits using the substitution formula u=g(x).
V
Vertex (of a parabola)
Quadratics
The turning point of a parabola y=ax²+bx+c — the minimum (a>0) or maximum (a<0). Found by completing the square: y=a(x+p)²+q gives vertex (−p, q). Also found using x=−b/(2a) for the x-coordinate, then substituting for y. The axis of symmetry passes through the vertex.
W
Well-Defined Function
Functions
A function is well-defined if every element of the domain maps to exactly one element — the output is unambiguous. For example, f(x)=√x is well-defined if we specify the positive square root. Inverse functions may not be well-defined unless the original function is one-to-one (i.e. domain is restricted if necessary).