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Tawjihiya Kuwait Scientific Stream Core Formulas and Principles

Essential theorems, equations, and foundational formulas across Pure Mathematics, Physics, and Chemistry for the Kuwait General Secondary Certificate.

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Fundamental Theorem of Calculus (Part 1 and 2)
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1
Fundamental Theorem of Calculus (Part 1 and 2)
If f is continuous on [a, b], then d/dx [integral from a to x of f(t) dt] = f(x), and integral from a to b of f(x) dx = F(b) - F(a) where F'(x) = f(x).
2
Derivative of Trigonometric Functions (sin, cos, tan)
d/dx [sin(x)] = cos(x); d/dx [cos(x)] = -sin(x); d/dx [tan(x)] = sec^2(x).
3
Chain Rule for Composite Functions
d/dx [f(g(x))] = f'(g(x)) g'(x); in Leibniz notation: dy/dx = (dy/du) (du/dx).
4
Product Rule and Quotient Rule of Differentiation
Product: (u v)' = u' v + u v'; Quotient: (u / v)' = (u' v - u * v') / v^2.
5
Integration by Parts Formula
integral of u dv = u * v - integral of v du, derived from the product rule of differentiation.
6
Area Between Two Continuous Curves y = f(x) and y = g(x)
Area = integral from a to b of [f(x) - g(x)] dx, where f(x) >= g(x) on the interval [a, b].
7
Volume of Revolution (Disk Method)
V = pi * integral from a to b of [R(x)]^2 dx for rotation around the x-axis.
8
Newton's Second Law of Motion
Sigma F = m * a = dp/dt; net external force equals mass times acceleration or time rate of change of momentum.
9
Work-Energy Theorem
W_net = Delta KE = 0.5 m v_f^2 - 0.5 m v_i^2; net work done on an object equals its change in kinetic energy.
10
Conservation of Linear Momentum
Sigma p_initial = Sigma p_final in a closed system isolated from external unbalanced forces: m1v1i + m2v2i = m1v1f + m2v2f.
11
Coulomb's Law of Electrostatic Force
F = k_e |q1 q2| / r^2, where k_e = 8.99 10^9 Nm^2/C^2 and r is separation distance.
12
Ohm's Law and Electrical Power Equations
V = I R; Electrical Power P = V I = I^2 * R = V^2 / R.
13
Equivalent Resistance in Series vs Parallel Circuits
Series: R_eq = R1 + R2 + ... + Rn; Parallel: 1 / R_eq = 1 / R1 + 1 / R2 + ... + 1 / Rn.
14
Faraday's Law of Electromagnetic Induction
emf = -N * (Delta Phi_B / Delta t); induced electromotive force equals negative time rate of change of magnetic flux through N loops.
15
Lenz's Law
The direction of an induced current always opposes the change in magnetic flux that produced it (accounting for the negative sign in Faraday's Law).
16
Snell's Law of Wave Refraction
n1 sin(theta1) = n2 sin(theta2); critical angle for total internal reflection: sin(theta_c) = n2 / n1 (when n1 > n2).
17
Ideal Gas Law Equation of State
P V = n R T, where R = 0.0821 Latm/(molK) or 8.314 J/(molK), with temperature in Kelvin.
18
Le Chatelier's Principle
When a chemical system at dynamic equilibrium experiences a stress (change in concentration, temperature, or pressure), the system shifts in the direction that counteracts the applied stress.
19
Arrhenius Equation for Reaction Rate Constant
k = A e^(-E_a / (R T)); increasing temperature or adding a catalyst (lowering activation energy E_a) exponentially increases rate constant k.
20
Definition of pH and pOH in Aqueous Solutions
pH = -log10[H3O+]; pOH = -log10[OH-]; at 25 degrees Celsius, pH + pOH = 14.00, and Kw = [H3O+][OH-] = 1.0 * 10^(-14).
21
Henderson-Hasselbalch Equation for Buffer Solutions
pH = pKa + log10([Conjugate Base A-] / [Weak Acid HA]); buffer capacity is maximized when pH = pKa.
22
Standard Cell Potential and Gibbs Free Energy Relation
Delta G_standard = -n F E_cell_standard; a positive standard cell potential corresponds to a negative Delta G and spontaneous reaction.
23
Markovnikov's Rule in Organic Electrophilic Addition
In the addition of HX (hydrogen halide) to an asymmetrical alkene, hydrogen bonds to the carbon already bearing more hydrogens, yielding the more stable carbocation intermediate.
24
Photoelectric Effect Equation (Einstein)
E_photon = h f = Phi (Work Function) + KE_max, where h = Planck's constant (6.626 10^(-34) Js) and KE_max = 0.5 m * v_max^2.
25
de Broglie Wavelength of Matter Waves
lambda = h / p = h / (m * v); establishes wave-particle duality for microscopic particles possessing mass and velocity.

General Secondary Certificate (Tawjihiya Kuwait) in Kuwait

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