Queensland Mathematical Methods Formula Book 2025

Mensuration Formulas

• Circumference of a circle:
  • $C = 2\text{π}r$
• Area of a circle:
  • $A = \text{π}r^2$
• Area of a parallelogram:
  • $A = bh$
• Area of a trapezium:
  • $A = \frac{1}{2}(a+b)h$
• Area of a triangle:
  • $A = \frac{1}{2}bh$
• Total surface area of a cone:
  • $S = \text{π}rs + \text{π}r^2$
• Total surface area of a cylinder:
  • $S = 2\text{π}rh + 2\text{π}r^2$
• Surface area of a sphere:
  • $S = 4\text{π}r^2$
• Volume of a cone:
  • $V = \frac{1}{3}\text{π}r^2h$
• Volume of a cylinder:
  • $V = \text{π}r^2h$
• Volume of a prism:
  • $V = Ah$
• Volume of a pyramid:
  • $V = \frac{1}{3}Ah$
• Volume of a sphere:
  • $V = \frac{4}{3}\text{π}r^3$

Graph Equations

• Quadratic equations:
  • $y = a(x-h)^2 + k$
  • $y = a(x-x_1)(x-x_2)$
• Cubic equations:
  • $y = a(x-h)^3 + k$
  • $y = a(x-x_1)(x-x_2)(x-x_3)$
• Circle equation:
  • $(x-h)^2 + (y-k)^2 = r^2$
• Square root equation:
  • $y = a∑−∑x-h + k$
• Reciprocal equation:
  • $y = \frac{a}{x-h} + k$
• Exponential equation:
  • $y = r^{x-h} + k$
  • Note: The condition $r > 0$ applies.
• Logarithmic equation:
  • $y = a∑−∑\text{log}_a(x-h) + k$
  • Note: The condition $a > 1$ applies.
• Trigonometric equations:
  • $y = a∑−∑\text{sin}(b(x-h)) + k$
  • $y = a∑−∑\text{cos}(b(x-h)) + k$

Logarithms and Exponents

• Relationship between exponents and logarithms:
  • $\text{log}_a(b) = x ⇔ a^x = b$
• Logarithmic laws and definitions:
  • Addition law: $\text{log}_a(x) + \text{log}_a(y) = \text{log}_a(xy)$
  • Power law: $\text{log}_a(x^n) = n\text{log}_a(x)$
  • Subtraction law: $\text{log}_a(x) - \text{log}_a(y) = \text{log}_a∑−∑\frac{x}{y}$
  • Change of base law: $\text{log}_a(x) = \frac{\text{log}_b(x)}{\text{log}_b(a)}$
  • Logarithm of one: $\text{log}_a(1) = 0$
  • Logarithm of the base: $\text{log}_a(a) = 1$

Calculus: Derivatives and Integrals

• First principles definition of a derivative:
  • $f'(x) = \text{lim}_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Limit of sums (Integration approximation):
  • \text∫_a^b f(x) dx ≈ \text∑ f(x_i) δx
• Fundamental Theorem of Calculus:
  • \text∫_a^b f(x) dx = F(b) - F(a)
• Linearity of the derivative:
  • $\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$
• Linearity of the integral:
  • \text∫ [f(x) + g(x)] dx = \text∫ f(x) dx + \text∫ g(x) dx
• Constant multiple rule for integrals:
  • \text∫ kf(x) dx = k\text∫ f(x) dx
• Power rule:
  • Derivative: $\frac{d}{dx}x^n = nx^{n-1}$
  • Integral: \text∫ x^n dx = \frac{x^{n+1}}{n+1} + c (where $n ≠ -1$)
• Exponential functions:
  • Derivative of the natural exponential: $\frac{d}{dx}e^x = e^x$
  • Derivative of a composite exponential: $\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$
  • Integral: \text∫ e^x dx = e^x + c
• Logarithmic functions:
  • Derivative of natural log: $\frac{d}{dx}\text{ln}(x) = \frac{1}{x}$
  • Derivative of log of a function: $\frac{d}{dx}\text{ln}(f(x)) = \frac{f'(x)}{f(x)}$
  • Integral: \text∫ \frac{1}{x} dx = \text{ln}(x) + c (for $x > 0$)
• Trigonometric functions (Sine):
  • Derivative: $\frac{d}{dx}\text{sin}(x) = \text{cos}(x)$
  • Derivative of composite: $\frac{d}{dx}\text{sin}(f(x)) = f'(x)\text{cos}(f(x))$
  • Integral: \text∫ \text{sin}(x) dx = -\text{cos}(x) + c
• Trigonometric functions (Cosine):
  • Derivative: $\frac{d}{dx}\text{cos}(x) = -\text{sin}(x)$
  • Derivative of composite: $\frac{d}{dx}\text{cos}(f(x)) = -f'(x)\text{sin}(f(x))$
  • Integral: \text∫ \text{cos}(x) dx = \text{sin}(x) + c

Calculus Rules and Numerical Integration

• Chain rule:
  • If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
• Product rule:
  • $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$
• Quotient rule:
  • $\frac{d}{dx}∑−∑\frac{u}{v} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$
• Trapezoidal rule:
  • \text∫_a^b f(x) dx ≈ \frac{w}{2} [f(x_0) + 2(f(x_1) + f(x_2) + f(x_3) + … + f(x_{n-1})) + f(x_n)]
  • Where the step width is defined as $w = \frac{b-a}{n}$

Trigonometry Rules and Identities

• Cosine rule:
  • $c^2 = a^2 + b^2 - 2ab\text{cos}(C)$
• Sine rule:
  • $\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}$
• Area of a triangle (Trigonometric):
  • $\text{Area} = \frac{1}{2}bc\text{sin}(A)$
• Pythagorean identity:
  • $\text{sin}^2(A) + \text{cos}^2(A) = 1$

Statistics and Probability

• Binomial theorem:
  • $(x+y)^n = x^n + \binom{n}{1}x^{n-1}y + … + \binom{n}{r}x^{n-r}y^r + … + y^n$
• Binomial probability:
  • $P(X=r) = \binom{n}{r}p^r(1-p)^{n-r}$
• Discrete random variable X:
  • Mean: $E(X) = ∑ p_i x_i = μ$
  • Variance: $\text{Var}(X) = ∑ p_i(x_i - μ)^2$
  • Standard deviation: $\text{SD} = ∑−∑\text{Var}(X)$
• Continuous random variable X:
  • Mean: E(X) = μ = \text∫_{-∞}^{∞} x p(x) dx
  • Variance: \text{Var}(X) = σ^2 = \text∫_{-∞}^{∞} (x-μ)^2 p(x) dx
• Bernoulli distribution:
  • Mean: $p$
  • Variance: $p(1-p)$
• Binomial distribution:
  • Mean: $np$
  • Variance: $np(1-p)$
• Sample proportion:
  • Mean: $μ = p$
  • Standard deviation: $σ = ∑−∑\frac{p(1-p)}{n}$
• Approximate confidence interval for p:
  • $\text−∑\text{p̂} - z∑−∑\frac{\text{p̂}(1-\text{p̂})}{n}, \text{p̂} + z∑−∑\frac{\text{p̂}(1-\text{p̂})}{n}\text−∑$
• Approximate margin of error:
  • $E = z∑−∑\frac{\text{p̂}(1-\text{p̂})}{n}$
• Probability definitions:
  • Complementary probability: $P(A') = 1 - P(A)$
  • General addition rule: $P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$
  • Probability of independent events: $P(A ∩ B) = P(A)P(B)$
  • Conditional probability: $P(A ∩ B) = P(A|B)P(B)$ or $P(A|B) = \frac{P(A ∩ B)}{P(B)}$