Formula Sheet - Ultimate Student Toolkit

Mathematics

Algebra

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

For equation $ax^2 + bx + c = 0$

Binomial Theorem

$$(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$$

Expansion of a binomial raised to a power

Logarithm Rules

$$\log_a(xy) = \log_a(x) + \log_a(y)$$ $$\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)$$ $$\log_a(x^n) = n\log_a(x)$$

Basic logarithm properties

Arithmetic Sequence

$$a_n = a_1 + (n-1)d$$ $$S_n = \frac{n}{2}(a_1 + a_n)$$

$a_n$ is the nth term, $d$ is common difference, $S_n$ is sum of n terms

Calculus

Derivative Rules

$$\frac{d}{dx}(x^n) = nx^{n-1}$$ $$\frac{d}{dx}(e^x) = e^x$$ $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$

Basic derivative rules

Integration Rules

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C, n \neq -1$$ $$\int e^x dx = e^x + C$$ $$\int \frac{1}{x} dx = \ln|x| + C$$

Basic integration rules

Chain Rule

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Derivative of composite functions

Integration by Parts

$$\int u(x)v'(x)dx = u(x)v(x) - \int v(x)u'(x)dx$$

Where $u(x)$ and $v(x)$ are functions

Geometry

Circle

$$A = \pi r^2$$ $$C = 2\pi r$$

$A$ is area, $C$ is circumference, $r$ is radius

Sphere

$$V = \frac{4}{3}\pi r^3$$ $$A = 4\pi r^2$$

$V$ is volume, $A$ is surface area, $r$ is radius

Pythagorean Theorem

$$a^2 + b^2 = c^2$$

In a right triangle, $c$ is hypotenuse, $a$ and $b$ are legs

Triangle Area

$$A = \frac{1}{2}bh$$ $$A = \frac{1}{2}ab\sin(C)$$

$b$ is base, $h$ is height, $C$ is included angle

Physics

Mechanics

Newton's Second Law

$$F = ma$$

$F$ is force, $m$ is mass, $a$ is acceleration

Kinematic Equations

$$v = v_0 + at$$ $$x = x_0 + v_0t + \frac{1}{2}at^2$$ $$v^2 = v_0^2 + 2a(x - x_0)$$

For constant acceleration

Work and Energy

$$W = Fd\cos\theta$$ $$KE = \frac{1}{2}mv^2$$ $$PE = mgh$$

$W$ is work, $KE$ is kinetic energy, $PE$ is potential energy

Momentum

$$p = mv$$ $$F = \frac{dp}{dt}$$

$p$ is momentum, $m$ is mass, $v$ is velocity

Electricity & Magnetism

Ohm's Law

$$V = IR$$

$V$ is voltage, $I$ is current, $R$ is resistance

Coulomb's Law

$$F = k\frac{q_1q_2}{r^2}$$

$F$ is force, $q$ are charges, $r$ is distance, $k$ is constant

Electric Power

$$P = VI$$ $$P = I^2R$$ $$P = \frac{V^2}{R}$$

$P$ is power, $V$ is voltage, $I$ is current, $R$ is resistance

Magnetic Field

$$F = qvB\sin\theta$$

$F$ is force, $q$ is charge, $v$ is velocity, $B$ is magnetic field

Thermodynamics

Ideal Gas Law

$$PV = nRT$$

$P$ is pressure, $V$ is volume, $n$ is moles, $R$ is gas constant, $T$ is temperature

Heat Transfer

$$Q = mc\Delta T$$

$Q$ is heat, $m$ is mass, $c$ is specific heat, $\Delta T$ is temperature change

Chemistry

General Chemistry

Molarity

$$M = \frac{\text{moles of solute}}{\text{liters of solution}}$$

$M$ is molarity in mol/L

pH Scale

$$pH = -\log[H^+]$$ $$pOH = -\log[OH^-]$$ $$pH + pOH = 14$$

At 25°C

Gas Laws

$$P_1V_1 = P_2V_2 \text{ (Boyle's Law)}$$ $$\frac{V_1}{T_1} = \frac{V_2}{T_2} \text{ (Charles' Law)}$$

At constant temperature and amount

Equilibrium Constant

$$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$$

For reaction $aA + bB \rightleftharpoons cC + dD$

Thermochemistry

Enthalpy Change

$$\Delta H = \sum \Delta H_f(\text{products}) - \sum \Delta H_f(\text{reactants})$$

$\Delta H_f$ is standard enthalpy of formation

Gibbs Free Energy

$$\Delta G = \Delta H - T\Delta S$$

$\Delta G$ is free energy change, $\Delta H$ is enthalpy change, $\Delta S$ is entropy change

Biology

Genetics

Hardy-Weinberg Equilibrium

$$p^2 + 2pq + q^2 = 1$$ $$p + q = 1$$

$p$ and $q$ are allele frequencies

Population Growth

$$\frac{dN}{dt} = rN(1-\frac{N}{K})$$

Logistic growth model, $r$ is growth rate, $K$ is carrying capacity

Engineering

Electrical Engineering

RC Circuit Time Constant

$$\tau = RC$$

$\tau$ is time constant, $R$ is resistance, $C$ is capacitance

Impedance

$$Z = R + jX$$

$Z$ is impedance, $R$ is resistance, $X$ is reactance

Mechanical Engineering

Stress and Strain

$$\sigma = \frac{F}{A}$$ $$\epsilon = \frac{\Delta L}{L}$$

$\sigma$ is stress, $\epsilon$ is strain, $F$ is force, $A$ is area

Fluid Mechanics

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2$$

Bernoulli's equation for fluid flow

Statistics & Probability

Descriptive Statistics

Mean, Median, Mode

$$\text{Mean} = \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$$

Measures of central tendency

Standard Deviation

$$\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$

Measure of dispersion in a dataset

Probability

Bayes' Theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Relates conditional probabilities

Normal Distribution

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$$

Probability density function for normal distribution