MATSEC IM Pure Mathematics cheat sheet

Remainder Theorem: dividing p(x) by (x − a) leaves remainder p(a). Factor Theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.

Division identity: p(x) = d(x)q(x) + r(x) with deg r < deg d; long division stops when the remainder has lower degree than the divisor.

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac))/(2a); the discriminant Δ = b² − 4ac gives two distinct real roots (Δ > 0), one repeated root (Δ = 0), no real roots (Δ < 0).

Root relations: for ax² + bx + c = 0 with roots α, β, α + β = −b/a and αβ = c/a.

Completing the square: ax² + bx + c = a(x + b/(2a))² + c − b²/(4a); the vertex of the parabola lies at x = −b/(2a).

Index laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Binomial theorem: (a + b)^n = Σ C(n,r) a^(n−r) b^r; the general term is C(n,r) a^(n−r) b^r with C(n,r) = n!/(r!(n−r)!).

Partial fractions: a proper rational function splits into A/(x − a) for distinct linear factors, A/(x − a) + B/(x − a)² for a repeated linear factor, and (Ax + B)/(irreducible quadratic) for a quadratic factor.

Absolute value: |x| < a with a > 0 means −a < x < a; |x| > a means x < −a or x > a.

Function notation f: A → B: A is the domain of allowed inputs, B is the codomain, and the range is the set of values actually taken by f(x).

Composite function: (f ∘ g)(x) = f(g(x)); it is defined only for x in the domain of g with g(x) in the domain of f.

A function has an inverse f⁻¹ only if it is one-to-one on its domain; then f⁻¹(f(x)) = x, and the domain and range of f and f⁻¹ are interchanged.

Graph tests: a vertical line meets the graph of a function at most once; a horizontal line meets the graph of a one-to-one function at most once.

Transformations: y = f(x) + a shifts the graph up by a; y = f(x − a) shifts it right by a; y = af(x) stretches it vertically by factor a; y = f(ax) stretches it horizontally by factor 1/a.

Reflections: y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects it in the y-axis; y = |f(x)| reflects the parts below the x-axis above it.

Symmetry: f is even if f(−x) = f(x), giving symmetry about the y-axis; f is odd if f(−x) = −f(x), giving rotational symmetry about the origin.

Rational functions: vertical asymptotes occur where the denominator vanishes after cancelling common factors; horizontal asymptotes are found from the limit of f(x) as x → ±∞.

Exponential y = a^x requires a > 0, a ≠ 1; it passes through (0, 1), has the x-axis as a horizontal asymptote, and is increasing if a > 1, decreasing if 0 < a < 1.

Logarithm definition: log_a x = y means a^y = x; log_a x is defined only for x > 0 with a > 0, a ≠ 1.

Log laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^n) = n log_a x; log_a 1 = 0; log_a a = 1.

Inverse pair: a^(log_a x) = x for x > 0, and log_a(a^x) = x for all real x; the graphs of y = a^x and y = log_a x are reflections in the line y = x.

Change of base: log_a x = log_b x / log_b a; in particular log_a x = ln x / ln a and log_a x = log₁₀ x / log₁₀ a.

The number e ≈ 2.71828; y = e^x is its own derivative, and ln x = log_e x satisfies ln 1 = 0 and ln e = 1.

Solving a^x = b: take logarithms of both sides to obtain x = log b / log a; for example 3^x = 7 gives x = ln 7 / ln 3.

Growth and decay: N = N₀e^(kt) with k > 0 for growth and k < 0 for decay; if N = N₀e^(−λt), the half-life is t = (ln 2)/λ.

Arithmetic sequence: u_n = a + (n − 1)d; sum S_n = (n/2)(2a + (n − 1)d) = (n/2)(a + l), where l is the last term.

Geometric sequence: u_n = ar^(n−1); sum S_n = a(1 − r^n)/(1 − r) for r ≠ 1, and S_n = na when r = 1.

Sum to infinity: a geometric series converges only if |r| < 1, in which case S_∞ = a/(1 − r); if |r| ≥ 1 the sum to infinity does not exist.

Standard sums: Σ_{r=1}^{n} r = n(n + 1)/2; Σ r² = n(n + 1)(2n + 1)/6; Σ r³ = [n(n + 1)/2]².

Binomial expansion for positive integer n: (1 + x)^n = 1 + nx + n(n−1)x²/2! + ... + x^n, terminating after n + 1 terms and valid for all x.

For negative or non-integer n the expansion of (1 + x)^n is infinite and valid only for |x| < 1; the coefficient of x^r is n(n−1)...(n−r+1)/r!.

Recurrence relation u_{n+1} = f(u_n) generates a sequence from a starting value; a sequence is increasing if u_{n+1} > u_n for all n, and periodic if terms repeat in a cycle.

Sigma notation: Σ_{r=m}^{n} u_r means u_m + u_{m+1} + ... + u_n; constant factors come outside the sum and sums split term by term.

Radian measure: π radians = 180°, so 1 radian = 180/π degrees; convert degrees to radians by multiplying by π/180.

Arc and sector with θ in radians: arc length s = rθ; sector area A = ½r²θ; segment area = sector area − triangle area.

Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ; tan θ = sin θ / cos θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = √2/2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A; triangle area = ½ab sin C.

Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).

Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; tan 2A = 2 tan A/(1 − tan²A).

Graphs: y = sin x and y = cos x have period 2π and amplitude 1; y = tan x has period π and vertical asymptotes at x = π/2 + nπ.

CAST: quadrant 1 has all ratios positive, quadrant 2 only sine, quadrant 3 only tangent, quadrant 4 only cosine; use the reference angle to list every solution in a given interval.

Distance and midpoint: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient m = (y₂ − y₁)/(x₂ − x₁); line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁); slope-intercept form y = mx + c.

Parallel lines have equal gradients (m₁ = m₂); perpendicular lines satisfy m₁m₂ = −1.

Circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r²; the expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

A tangent to a circle is perpendicular to the radius at the point of contact, so the tangent gradient equals −1 divided by the radius gradient.

Distance from the point (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²).

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Intersection of two lines: solve their equations simultaneously; the coordinates found satisfy both equations at once.

Component form: a = x i + y j + z k; magnitude |a| = √(x² + y² + z²); the unit vector in the direction of a is a/|a|.

Position vector of point A relative to origin O is OA; the vector from A to B is AB = b − a, and |AB| gives the distance between the points.

Scalar (dot) product: a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂, where θ is the angle between the vectors.

Perpendicular vectors: a·b = 0 for non-zero a and b; the angle between vectors satisfies cos θ = (a·b)/(|a||b|).

Vector equation of a line: r = a + t d, where a is the position vector of a point on the line, d is a direction vector, and t is a real parameter.

Two lines are parallel if their direction vectors are scalar multiples; they intersect if the equations have a common solution, and in three dimensions non-parallel non-intersecting lines are skew.

Scalar projection of a on b is (a·b)/|b|; the vector projection of a on b is [(a·b)/|b|²] b.

Vector addition is componentwise; multiplying by a scalar λ changes length by |λ| and reverses direction when λ < 0.

Cartesian form: z = a + bi with a = Re z, b = Im z and i² = −1; two complex numbers are equal only if both real parts and imaginary parts match.

Modulus and argument: |z| = √(a² + b²); arg z is the angle from the positive real axis, with tan θ = b/a and the quadrant fixed by the signs of a and b.

Conjugate: z* = a − bi; z + z* = 2a, z − z* = 2bi, z z* = a² + b² = |z|², and 1/z = z*/|z|².

Polar form: z = r(cos θ + i sin θ) = r e^(iθ), where r = |z| and θ = arg z.

Multiplication and division in polar form: multiply the moduli and add the arguments; divide the moduli and subtract the arguments.

De Moivre: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ) for integer n; this evaluates powers and roots efficiently.

The n distinct nth roots of w = r(cos θ + i sin θ) are r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, ..., n − 1.

For equations with real coefficients, non-real roots occur in conjugate pairs; on an Argand diagram the equation |z − z₀| = r represents a circle of centre z₀ and radius r.

Derivative as a limit: f'(x) = lim_{h→0} [f(x + h) − f(x)]/h; it gives the gradient of the tangent and the instantaneous rate of change.

Power rule: d/dx (x^n) = n x^(n−1) for any real n where defined; d/dx (constant) = 0; d/dx (ax^n) = an x^(n−1).

Rules: product (uv)' = u'v + uv'; quotient (u/v)' = (u'v − uv')/v²; chain rule dy/dx = (dy/du)(du/dx).

Standard derivatives: d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x; d/dx e^x = e^x; d/dx ln x = 1/x; d/dx a^x = a^x ln a.

Stationary points satisfy f'(x) = 0; classify them by f''(x) > 0 (minimum) or f''(x) < 0 (maximum), or by the sign change of f'(x).

Tangent at x = a: y − f(a) = f'(a)(x − a); the normal gradient is −1/f'(a), so the normal is y − f(a) = −[1/f'(a)](x − a).

Monotonicity: f is increasing where f'(x) > 0 and decreasing where f'(x) < 0; points where f' changes sign are local maxima or minima.

Implicit differentiation: differentiate both sides with respect to x, remembering d/dx(y) = dy/dx by the chain rule, then solve for dy/dx.

Connected rates of change: if y depends on x and x depends on t, then dy/dt = (dy/dx)(dx/dt).

Indefinite integrals: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; always include the constant of integration.

Standard integrals: ∫e^x dx = e^x + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C; ∫a^x dx = a^x/ln a + C.

Definite integral: ∫_a^b f(x) dx = F(b) − F(a) where F' = f; it gives the signed area between the curve and the x-axis from a to b.

Area between curves: ∫_a^b (upper curve − lower curve) dx, where a and b are the x-coordinates of the intersection points.

Substitution: ∫f(g(x))g'(x) dx = ∫f(u) du with u = g(x); for definite integrals change the limits to u-values.

Integration by parts: ∫u dv = uv − ∫v du; choose u as the factor that simplifies when differentiated, often a polynomial, logarithm or inverse trigonometric function.

Volume of revolution about the x-axis: V = π∫_a^b y² dx; about the y-axis use V = π∫ x² dy.

Trapezium rule with n strips of width h = (b − a)/n: ∫a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁ + ... + y{n−1}) + y_n].

Area below the x-axis gives a negative definite integral; for total area split the interval at each root and add the absolute values of the separate integrals.

Remainder Theorem: dividing p(x) by (x − a) leaves remainder p(a). Factor Theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.

Division identity: p(x) = d(x)q(x) + r(x) with deg r < deg d; long division stops when the remainder has lower degree than the divisor.

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac))/(2a); the discriminant Δ = b² − 4ac gives two distinct real roots (Δ > 0), one repeated root (Δ = 0), no real roots (Δ < 0).

Root relations: for ax² + bx + c = 0 with roots α, β, α + β = −b/a and αβ = c/a.

Completing the square: ax² + bx + c = a(x + b/(2a))² + c − b²/(4a); the vertex of the parabola lies at x = −b/(2a).

Index laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Binomial theorem: (a + b)^n = Σ C(n,r) a^(n−r) b^r; the general term is C(n,r) a^(n−r) b^r with C(n,r) = n!/(r!(n−r)!).

Partial fractions: a proper rational function splits into A/(x − a) for distinct linear factors, A/(x − a) + B/(x − a)² for a repeated linear factor, and (Ax + B)/(irreducible quadratic) for a quadratic factor.

Absolute value: |x| < a with a > 0 means −a < x < a; |x| > a means x < −a or x > a.

Function notation f: A → B: A is the domain of allowed inputs, B is the codomain, and the range is the set of values actually taken by f(x).

Composite function: (f ∘ g)(x) = f(g(x)); it is defined only for x in the domain of g with g(x) in the domain of f.

A function has an inverse f⁻¹ only if it is one-to-one on its domain; then f⁻¹(f(x)) = x, and the domain and range of f and f⁻¹ are interchanged.

Graph tests: a vertical line meets the graph of a function at most once; a horizontal line meets the graph of a one-to-one function at most once.

Transformations: y = f(x) + a shifts the graph up by a; y = f(x − a) shifts it right by a; y = af(x) stretches it vertically by factor a; y = f(ax) stretches it horizontally by factor 1/a.

Reflections: y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects it in the y-axis; y = |f(x)| reflects the parts below the x-axis above it.

Symmetry: f is even if f(−x) = f(x), giving symmetry about the y-axis; f is odd if f(−x) = −f(x), giving rotational symmetry about the origin.

Rational functions: vertical asymptotes occur where the denominator vanishes after cancelling common factors; horizontal asymptotes are found from the limit of f(x) as x → ±∞.

Exponential y = a^x requires a > 0, a ≠ 1; it passes through (0, 1), has the x-axis as a horizontal asymptote, and is increasing if a > 1, decreasing if 0 < a < 1.

Logarithm definition: log_a x = y means a^y = x; log_a x is defined only for x > 0 with a > 0, a ≠ 1.

Log laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^n) = n log_a x; log_a 1 = 0; log_a a = 1.

Inverse pair: a^(log_a x) = x for x > 0, and log_a(a^x) = x for all real x; the graphs of y = a^x and y = log_a x are reflections in the line y = x.

Change of base: log_a x = log_b x / log_b a; in particular log_a x = ln x / ln a and log_a x = log₁₀ x / log₁₀ a.

The number e ≈ 2.71828; y = e^x is its own derivative, and ln x = log_e x satisfies ln 1 = 0 and ln e = 1.

Solving a^x = b: take logarithms of both sides to obtain x = log b / log a; for example 3^x = 7 gives x = ln 7 / ln 3.

Growth and decay: N = N₀e^(kt) with k > 0 for growth and k < 0 for decay; if N = N₀e^(−λt), the half-life is t = (ln 2)/λ.

Arithmetic sequence: u_n = a + (n − 1)d; sum S_n = (n/2)(2a + (n − 1)d) = (n/2)(a + l), where l is the last term.

Geometric sequence: u_n = ar^(n−1); sum S_n = a(1 − r^n)/(1 − r) for r ≠ 1, and S_n = na when r = 1.

Sum to infinity: a geometric series converges only if |r| < 1, in which case S_∞ = a/(1 − r); if |r| ≥ 1 the sum to infinity does not exist.

Standard sums: Σ_{r=1}^{n} r = n(n + 1)/2; Σ r² = n(n + 1)(2n + 1)/6; Σ r³ = [n(n + 1)/2]².

Binomial expansion for positive integer n: (1 + x)^n = 1 + nx + n(n−1)x²/2! + ... + x^n, terminating after n + 1 terms and valid for all x.

For negative or non-integer n the expansion of (1 + x)^n is infinite and valid only for |x| < 1; the coefficient of x^r is n(n−1)...(n−r+1)/r!.

Recurrence relation u_{n+1} = f(u_n) generates a sequence from a starting value; a sequence is increasing if u_{n+1} > u_n for all n, and periodic if terms repeat in a cycle.

Sigma notation: Σ_{r=m}^{n} u_r means u_m + u_{m+1} + ... + u_n; constant factors come outside the sum and sums split term by term.

Radian measure: π radians = 180°, so 1 radian = 180/π degrees; convert degrees to radians by multiplying by π/180.

Arc and sector with θ in radians: arc length s = rθ; sector area A = ½r²θ; segment area = sector area − triangle area.

Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ; tan θ = sin θ / cos θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = √2/2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A; triangle area = ½ab sin C.

Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).

Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; tan 2A = 2 tan A/(1 − tan²A).

Graphs: y = sin x and y = cos x have period 2π and amplitude 1; y = tan x has period π and vertical asymptotes at x = π/2 + nπ.

CAST: quadrant 1 has all ratios positive, quadrant 2 only sine, quadrant 3 only tangent, quadrant 4 only cosine; use the reference angle to list every solution in a given interval.

Distance and midpoint: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient m = (y₂ − y₁)/(x₂ − x₁); line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁); slope-intercept form y = mx + c.

Parallel lines have equal gradients (m₁ = m₂); perpendicular lines satisfy m₁m₂ = −1.

Circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r²; the expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

A tangent to a circle is perpendicular to the radius at the point of contact, so the tangent gradient equals −1 divided by the radius gradient.

Distance from the point (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²).

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Intersection of two lines: solve their equations simultaneously; the coordinates found satisfy both equations at once.

Component form: a = x i + y j + z k; magnitude |a| = √(x² + y² + z²); the unit vector in the direction of a is a/|a|.

Position vector of point A relative to origin O is OA; the vector from A to B is AB = b − a, and |AB| gives the distance between the points.

Scalar (dot) product: a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂, where θ is the angle between the vectors.

Perpendicular vectors: a·b = 0 for non-zero a and b; the angle between vectors satisfies cos θ = (a·b)/(|a||b|).

Vector equation of a line: r = a + t d, where a is the position vector of a point on the line, d is a direction vector, and t is a real parameter.

Two lines are parallel if their direction vectors are scalar multiples; they intersect if the equations have a common solution, and in three dimensions non-parallel non-intersecting lines are skew.

Scalar projection of a on b is (a·b)/|b|; the vector projection of a on b is [(a·b)/|b|²] b.

Vector addition is componentwise; multiplying by a scalar λ changes length by |λ| and reverses direction when λ < 0.

Cartesian form: z = a + bi with a = Re z, b = Im z and i² = −1; two complex numbers are equal only if both real parts and imaginary parts match.

Modulus and argument: |z| = √(a² + b²); arg z is the angle from the positive real axis, with tan θ = b/a and the quadrant fixed by the signs of a and b.

Conjugate: z* = a − bi; z + z* = 2a, z − z* = 2bi, z z* = a² + b² = |z|², and 1/z = z*/|z|².

Polar form: z = r(cos θ + i sin θ) = r e^(iθ), where r = |z| and θ = arg z.

Multiplication and division in polar form: multiply the moduli and add the arguments; divide the moduli and subtract the arguments.

De Moivre: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ) for integer n; this evaluates powers and roots efficiently.

The n distinct nth roots of w = r(cos θ + i sin θ) are r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, ..., n − 1.

For equations with real coefficients, non-real roots occur in conjugate pairs; on an Argand diagram the equation |z − z₀| = r represents a circle of centre z₀ and radius r.

Derivative as a limit: f'(x) = lim_{h→0} [f(x + h) − f(x)]/h; it gives the gradient of the tangent and the instantaneous rate of change.

Power rule: d/dx (x^n) = n x^(n−1) for any real n where defined; d/dx (constant) = 0; d/dx (ax^n) = an x^(n−1).

Rules: product (uv)' = u'v + uv'; quotient (u/v)' = (u'v − uv')/v²; chain rule dy/dx = (dy/du)(du/dx).

Standard derivatives: d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x; d/dx e^x = e^x; d/dx ln x = 1/x; d/dx a^x = a^x ln a.

Stationary points satisfy f'(x) = 0; classify them by f''(x) > 0 (minimum) or f''(x) < 0 (maximum), or by the sign change of f'(x).

Tangent at x = a: y − f(a) = f'(a)(x − a); the normal gradient is −1/f'(a), so the normal is y − f(a) = −[1/f'(a)](x − a).

Monotonicity: f is increasing where f'(x) > 0 and decreasing where f'(x) < 0; points where f' changes sign are local maxima or minima.

Implicit differentiation: differentiate both sides with respect to x, remembering d/dx(y) = dy/dx by the chain rule, then solve for dy/dx.

Connected rates of change: if y depends on x and x depends on t, then dy/dt = (dy/dx)(dx/dt).

Indefinite integrals: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; always include the constant of integration.

Standard integrals: ∫e^x dx = e^x + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C; ∫a^x dx = a^x/ln a + C.

Definite integral: ∫_a^b f(x) dx = F(b) − F(a) where F' = f; it gives the signed area between the curve and the x-axis from a to b.

Area between curves: ∫_a^b (upper curve − lower curve) dx, where a and b are the x-coordinates of the intersection points.

Substitution: ∫f(g(x))g'(x) dx = ∫f(u) du with u = g(x); for definite integrals change the limits to u-values.

Integration by parts: ∫u dv = uv − ∫v du; choose u as the factor that simplifies when differentiated, often a polynomial, logarithm or inverse trigonometric function.

Volume of revolution about the x-axis: V = π∫_a^b y² dx; about the y-axis use V = π∫ x² dy.

Trapezium rule with n strips of width h = (b − a)/n: ∫a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁ + ... + y{n−1}) + y_n].

Area below the x-axis gives a negative definite integral; for total area split the interval at each root and add the absolute values of the separate integrals.

Remainder Theorem: dividing p(x) by (x − a) leaves remainder p(a). Factor Theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.

Division identity: p(x) = d(x)q(x) + r(x) with deg r < deg d; long division stops when the remainder has lower degree than the divisor.

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac))/(2a); the discriminant Δ = b² − 4ac gives two distinct real roots (Δ > 0), one repeated root (Δ = 0), no real roots (Δ < 0).

Root relations: for ax² + bx + c = 0 with roots α, β, α + β = −b/a and αβ = c/a.

Completing the square: ax² + bx + c = a(x + b/(2a))² + c − b²/(4a); the vertex of the parabola lies at x = −b/(2a).

Index laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Binomial theorem: (a + b)^n = Σ C(n,r) a^(n−r) b^r; the general term is C(n,r) a^(n−r) b^r with C(n,r) = n!/(r!(n−r)!).

Partial fractions: a proper rational function splits into A/(x − a) for distinct linear factors, A/(x − a) + B/(x − a)² for a repeated linear factor, and (Ax + B)/(irreducible quadratic) for a quadratic factor.

Absolute value: |x| < a with a > 0 means −a < x < a; |x| > a means x < −a or x > a.

Function notation f: A → B: A is the domain of allowed inputs, B is the codomain, and the range is the set of values actually taken by f(x).

Composite function: (f ∘ g)(x) = f(g(x)); it is defined only for x in the domain of g with g(x) in the domain of f.

A function has an inverse f⁻¹ only if it is one-to-one on its domain; then f⁻¹(f(x)) = x, and the domain and range of f and f⁻¹ are interchanged.

Graph tests: a vertical line meets the graph of a function at most once; a horizontal line meets the graph of a one-to-one function at most once.

Transformations: y = f(x) + a shifts the graph up by a; y = f(x − a) shifts it right by a; y = af(x) stretches it vertically by factor a; y = f(ax) stretches it horizontally by factor 1/a.

Reflections: y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects it in the y-axis; y = |f(x)| reflects the parts below the x-axis above it.

Symmetry: f is even if f(−x) = f(x), giving symmetry about the y-axis; f is odd if f(−x) = −f(x), giving rotational symmetry about the origin.

Rational functions: vertical asymptotes occur where the denominator vanishes after cancelling common factors; horizontal asymptotes are found from the limit of f(x) as x → ±∞.

Exponential y = a^x requires a > 0, a ≠ 1; it passes through (0, 1), has the x-axis as a horizontal asymptote, and is increasing if a > 1, decreasing if 0 < a < 1.

Logarithm definition: log_a x = y means a^y = x; log_a x is defined only for x > 0 with a > 0, a ≠ 1.

Log laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^n) = n log_a x; log_a 1 = 0; log_a a = 1.

Inverse pair: a^(log_a x) = x for x > 0, and log_a(a^x) = x for all real x; the graphs of y = a^x and y = log_a x are reflections in the line y = x.

Change of base: log_a x = log_b x / log_b a; in particular log_a x = ln x / ln a and log_a x = log₁₀ x / log₁₀ a.

The number e ≈ 2.71828; y = e^x is its own derivative, and ln x = log_e x satisfies ln 1 = 0 and ln e = 1.

Solving a^x = b: take logarithms of both sides to obtain x = log b / log a; for example 3^x = 7 gives x = ln 7 / ln 3.

Growth and decay: N = N₀e^(kt) with k > 0 for growth and k < 0 for decay; if N = N₀e^(−λt), the half-life is t = (ln 2)/λ.

Arithmetic sequence: u_n = a + (n − 1)d; sum S_n = (n/2)(2a + (n − 1)d) = (n/2)(a + l), where l is the last term.

Geometric sequence: u_n = ar^(n−1); sum S_n = a(1 − r^n)/(1 − r) for r ≠ 1, and S_n = na when r = 1.

Sum to infinity: a geometric series converges only if |r| < 1, in which case S_∞ = a/(1 − r); if |r| ≥ 1 the sum to infinity does not exist.

Standard sums: Σ_{r=1}^{n} r = n(n + 1)/2; Σ r² = n(n + 1)(2n + 1)/6; Σ r³ = [n(n + 1)/2]².

Binomial expansion for positive integer n: (1 + x)^n = 1 + nx + n(n−1)x²/2! + ... + x^n, terminating after n + 1 terms and valid for all x.

For negative or non-integer n the expansion of (1 + x)^n is infinite and valid only for |x| < 1; the coefficient of x^r is n(n−1)...(n−r+1)/r!.

Recurrence relation u_{n+1} = f(u_n) generates a sequence from a starting value; a sequence is increasing if u_{n+1} > u_n for all n, and periodic if terms repeat in a cycle.

Sigma notation: Σ_{r=m}^{n} u_r means u_m + u_{m+1} + ... + u_n; constant factors come outside the sum and sums split term by term.

Radian measure: π radians = 180°, so 1 radian = 180/π degrees; convert degrees to radians by multiplying by π/180.

Arc and sector with θ in radians: arc length s = rθ; sector area A = ½r²θ; segment area = sector area − triangle area.

Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ; tan θ = sin θ / cos θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = √2/2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² − 2bc cos A; triangle area = ½ab sin C.

Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).

Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1; tan 2A = 2 tan A/(1 − tan²A).

Graphs: y = sin x and y = cos x have period 2π and amplitude 1; y = tan x has period π and vertical asymptotes at x = π/2 + nπ.

CAST: quadrant 1 has all ratios positive, quadrant 2 only sine, quadrant 3 only tangent, quadrant 4 only cosine; use the reference angle to list every solution in a given interval.

Distance and midpoint: d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient m = (y₂ − y₁)/(x₂ − x₁); line through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁); slope-intercept form y = mx + c.

Parallel lines have equal gradients (m₁ = m₂); perpendicular lines satisfy m₁m₂ = −1.

Circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r²; the expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

A tangent to a circle is perpendicular to the radius at the point of contact, so the tangent gradient equals −1 divided by the radius gradient.

Distance from the point (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²).

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Intersection of two lines: solve their equations simultaneously; the coordinates found satisfy both equations at once.

Component form: a = x i + y j + z k; magnitude |a| = √(x² + y² + z²); the unit vector in the direction of a is a/|a|.

Position vector of point A relative to origin O is OA; the vector from A to B is AB = b − a, and |AB| gives the distance between the points.

Scalar (dot) product: a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂, where θ is the angle between the vectors.

Perpendicular vectors: a·b = 0 for non-zero a and b; the angle between vectors satisfies cos θ = (a·b)/(|a||b|).

Vector equation of a line: r = a + t d, where a is the position vector of a point on the line, d is a direction vector, and t is a real parameter.

Two lines are parallel if their direction vectors are scalar multiples; they intersect if the equations have a common solution, and in three dimensions non-parallel non-intersecting lines are skew.

Scalar projection of a on b is (a·b)/|b|; the vector projection of a on b is [(a·b)/|b|²] b.

Vector addition is componentwise; multiplying by a scalar λ changes length by |λ| and reverses direction when λ < 0.

Cartesian form: z = a + bi with a = Re z, b = Im z and i² = −1; two complex numbers are equal only if both real parts and imaginary parts match.

Modulus and argument: |z| = √(a² + b²); arg z is the angle from the positive real axis, with tan θ = b/a and the quadrant fixed by the signs of a and b.

Conjugate: z* = a − bi; z + z* = 2a, z − z* = 2bi, z z* = a² + b² = |z|², and 1/z = z*/|z|².

Polar form: z = r(cos θ + i sin θ) = r e^(iθ), where r = |z| and θ = arg z.

Multiplication and division in polar form: multiply the moduli and add the arguments; divide the moduli and subtract the arguments.

De Moivre: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ) for integer n; this evaluates powers and roots efficiently.

The n distinct nth roots of w = r(cos θ + i sin θ) are r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, ..., n − 1.

For equations with real coefficients, non-real roots occur in conjugate pairs; on an Argand diagram the equation |z − z₀| = r represents a circle of centre z₀ and radius r.

Derivative as a limit: f'(x) = lim_{h→0} [f(x + h) − f(x)]/h; it gives the gradient of the tangent and the instantaneous rate of change.

Power rule: d/dx (x^n) = n x^(n−1) for any real n where defined; d/dx (constant) = 0; d/dx (ax^n) = an x^(n−1).

Rules: product (uv)' = u'v + uv'; quotient (u/v)' = (u'v − uv')/v²; chain rule dy/dx = (dy/du)(du/dx).

Standard derivatives: d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x; d/dx e^x = e^x; d/dx ln x = 1/x; d/dx a^x = a^x ln a.

Stationary points satisfy f'(x) = 0; classify them by f''(x) > 0 (minimum) or f''(x) < 0 (maximum), or by the sign change of f'(x).

Tangent at x = a: y − f(a) = f'(a)(x − a); the normal gradient is −1/f'(a), so the normal is y − f(a) = −[1/f'(a)](x − a).

Monotonicity: f is increasing where f'(x) > 0 and decreasing where f'(x) < 0; points where f' changes sign are local maxima or minima.

Implicit differentiation: differentiate both sides with respect to x, remembering d/dx(y) = dy/dx by the chain rule, then solve for dy/dx.

Connected rates of change: if y depends on x and x depends on t, then dy/dt = (dy/dx)(dx/dt).

Indefinite integrals: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; always include the constant of integration.

Standard integrals: ∫e^x dx = e^x + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C; ∫a^x dx = a^x/ln a + C.

Definite integral: ∫_a^b f(x) dx = F(b) − F(a) where F' = f; it gives the signed area between the curve and the x-axis from a to b.

Area between curves: ∫_a^b (upper curve − lower curve) dx, where a and b are the x-coordinates of the intersection points.

Substitution: ∫f(g(x))g'(x) dx = ∫f(u) du with u = g(x); for definite integrals change the limits to u-values.

Integration by parts: ∫u dv = uv − ∫v du; choose u as the factor that simplifies when differentiated, often a polynomial, logarithm or inverse trigonometric function.

Volume of revolution about the x-axis: V = π∫_a^b y² dx; about the y-axis use V = π∫ x² dy.

Trapezium rule with n strips of width h = (b − a)/n: ∫a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁ + ... + y{n−1}) + y_n].

Area below the x-axis gives a negative definite integral; for total area split the interval at each root and add the absolute values of the separate integrals.