CAPE Pure Maths Unit 1 cheat sheet

Remainder Theorem: If a polynomial f(x) is divided by (x − a), the remainder is f(a). Factor Theorem: (x − a) is a factor iff f(a) = 0.

For a quadratic ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 two distinct real, Δ = 0 one repeated real, Δ < 0 two complex conjugate roots.

Sum and product of roots of ax² + bx + c = 0: α + β = −b/a, αβ = c/a. For a cubic ax³ + bx² + cx + d = 0: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

Partial fractions: For a proper rational function, factor the denominator. Each distinct linear factor (ax + b) gives A/(ax + b); each repeated linear factor (ax + b)^k gives A₁/(ax + b) + ... + A_k/(ax + b)^k; each irreducible quadratic (ax² + bx + c) gives (Bx + C)/(ax² + bx + c).

Inequalities: Multiplying or dividing by a negative number reverses the inequality sign. For quadratic inequalities, find critical values and test intervals, or sketch the parabola.

Binomial theorem: (a + b)^n = Σ_{r=0}^n C(n, r) a^{n−r} b^r, where C(n, r) = n!/(r!(n−r)!). The general term is C(n, r) a^{n−r} b^r.

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; log_a b = (log_c b)/(log_c a). Also a^{log_a x} = x.

Exponential equations: If a^{f(x)} = a^{g(x)} with a > 0, a ≠ 1, then f(x) = g(x). If a^x = b, take logarithms: x = log b / log a.

Modulus: |x| = x if x ≥ 0, |x| = −x if x < 0. |x| < a (a > 0) ⇔ −a < x < a; |x| > a ⇔ x < −a or x > a.

A function f: A → B is one-to-one (injective) if f(a) = f(b) implies a = b; it is onto (surjective) if every element of B is the image of some element of A. A bijection is both.

The inverse function f⁻¹ exists iff f is a bijection. Its domain is the range of f and its range is the domain of f. The graphs of y = f(x) and y = f⁻¹(x) are reflections in the line y = x.

Composite function (f ∘ g)(x) = f(g(x)) is defined only when the range of g is contained in the domain of f.

Transformations: y = f(x) + a shifts up by a; y = f(x + a) shifts left by a; y = a f(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a; y = −f(x) reflects in x-axis; y = f(−x) reflects in y-axis.

A function is even if f(−x) = f(x) for all x (symmetric about y-axis); it is odd if f(−x) = −f(x) for all x (symmetric about origin).

Rational functions: For y = (ax + b)/(cx + d), the vertical asymptote is x = −d/c (if c ≠ 0) and the horizontal asymptote is y = a/c. The domain excludes x = −d/c.

Exponential and logarithmic graphs: y = e^x has domain ℝ, range (0, ∞), horizontal asymptote y = 0. y = ln x has domain (0, ∞), range ℝ, vertical asymptote x = 0. They are inverse functions.

Modulus graph: y = |f(x)| is obtained by reflecting the part of y = f(x) below the x-axis above it. y = f(|x|) is obtained by discarding the part for x < 0 and reflecting the part for x ≥ 0 in the y-axis.

Distance between points (x₁, y₁) and (x₂, y₂): d = √((x₂ − x₁)² + (y₂ − y₁)²). Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient of a line through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁). Equation: y − y₁ = m(x − x₁). Parallel lines have equal gradients; perpendicular lines have gradients m₁m₂ = −1.

Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r². Expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

Radians: π radians = 180°. Arc length s = rθ and sector area A = ½ r² θ, where θ is in radians.

Trigonometric identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Also 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 angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2tan θ/(1 − tan²θ).

General solutions: If sin θ = sin α, then θ = nπ + (−1)^n α; if cos θ = cos α, then θ = 2nπ ± α; if tan θ = tan α, then θ = nπ + α, for integer n.

The equation a cos θ + b sin θ = c can be written as R cos(θ − α) = c, where R = √(a² + b²) and tan α = b/a. Maximum value is R, minimum is −R.

Derivative from first principles: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.

Standard derivatives: d/dx (x^n) = n x^{n−1}; d/dx (e^x) = e^x; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec²x.

Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v². Chain rule: dy/dx = dy/du × du/dx.

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.

Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Standard integrals: ∫ x^n dx = x^{n+1}/(n+1) + C (n ≠ −1); ∫ (1/x) dx = ln|x| + C; ∫ e^x dx = e^x + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec²x dx = tan x + C.

Integration by substitution: choose u = g(x), then ∫ f(g(x)) g′(x) dx = ∫ f(u) du. For definite integrals, change the limits accordingly.

Integration by parts: ∫ u dv = uv − ∫ v du. Choose u according to LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

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

Tangents and normals: gradient of tangent at x = a is f′(a). Equation of tangent: y − f(a) = f′(a)(x − a). Normal gradient is −1/f′(a) (if f′(a) ≠ 0).

Stationary points occur where f′(x) = 0. Use the second derivative test: if f″(x) > 0, local minimum; if f″(x) < 0, local maximum; if f″(x) = 0, test inconclusive (use first derivative sign change).

Rates of change: if y = f(x) and x = g(t), then dy/dt = (dy/dx)(dx/dt). Related rates problems use this chain rule.

Small increments: Δy ≈ (dy/dx) Δx. Percentage error in y ≈ percentage error in x multiplied by (x/y)(dy/dx).

Area under a curve: A = ∫_a^b y dx for y ≥ 0. If the curve crosses the x-axis, integrate each section separately and take absolute values.

Area between two curves: A = ∫_a^b [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].

Kinematics: if displacement s(t), then velocity v = ds/dt and acceleration a = dv/dt = d²s/dt². Conversely, displacement is the integral of velocity.

Optimisation: express the quantity to be optimised as a function of one variable, find stationary points, and use the second derivative or endpoints to determine the maximum or minimum.