Lebanese University Faculty of Engineering Entrance Examination flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Discriminant of a quadratic

For ax² + bx + c = 0, the quantity Δ = b² − 4ac. Positive Δ gives two distinct real roots, zero gives one double root, negative gives no real roots.

Vieta's sum and product

For ax² + bx + c = 0 with roots x₁ and x₂, the sum x₁ + x₂ equals −b/a and the product x₁x₂ equals c/a, without solving the equation.

Change of logarithm base

log_a b equals ln b divided by ln a, or log b over log a in any common base. This lets you evaluate logarithms on a calculator that only offers base ten or natural logs.

Domain of a square root

The expression under an even root must be nonnegative. Solve the resulting inequality and intersect it with every other restriction coming from denominators or logarithms in the same function.

Sum of a geometric series

For first term u₁ and ratio q ≠ 1, the sum of n terms is u₁(1 − qⁿ)/(1 − q). If |q| < 1, the infinite sum converges to u₁/(1 − q).

Horizontal asymptote rule

If numerator and denominator of a rational function have equal degree, the horizontal asymptote is the ratio of leading coefficients. If the denominator degree is larger, the asymptote is y = 0.

Pythagorean identity

sin²θ + cos²θ = 1 for every angle θ. Dividing by cos²θ gives 1 + tan²θ = 1/cos²θ, and dividing by sin²θ gives 1 + cot²θ = 1/sin²θ.

Cosine rule

In any triangle, a² = b² + c² − 2bc·cos A, where A is the angle opposite side a. It determines a side from two sides and the included angle, or an angle from three sides.

Sine rule and circumradius

a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circle through the three vertices. Use it when a side is paired with its opposite angle.

Double-angle formulas

sin 2a = 2 sin a cos a, and cos 2a = cos²a − sin²a = 2cos²a − 1 = 1 − 2sin²a. Also tan 2a = 2 tan a/(1 − tan²a) where defined.

Dot product and perpendicularity

u·v = x₁x₂ + y₁y₂ = |u||v|cos θ. Two nonzero vectors are perpendicular exactly when their dot product is zero, and the angle follows from cos θ = u·v/(|u||v|).

Tangent line to a circle

Substitute the line equation into the circle equation to get a quadratic. The line is tangent when this quadratic has a zero discriminant, secant when positive, and does not meet the circle when negative.

Derivative as a limit

f′(x) is the limit as h tends to zero of [f(x + h) − f(x)]/h. It gives the slope of the tangent line and the instantaneous rate of change at x.

Chain rule

The derivative of f(g(x)) is f′(g(x))·g′(x). Differentiate the outer function at the inner value, then multiply by the derivative of the inner function.

L'Hôpital's rule

For a limit giving the indeterminate form 0/0 or ∞/∞, replace the ratio by the ratio of derivatives. Repeat if needed, provided the resulting limit exists.

First-derivative test

At a critical point where f′ = 0, if f′ changes from positive to negative the point is a local maximum; from negative to positive it is a local minimum.

Fundamental theorem of calculus

If F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a). It links areas under curves with antiderivatives.

Integration by parts

The rule ∫u dv = uv − ∫v du reverses the product rule. Choose u so that its derivative simplifies and dv so that it is easy to integrate.

Projectile range on level ground

For launch speed v₀ at angle θ with negligible air resistance, range equals v₀²sin 2θ/g. The range is greatest at 45°, and complementary angles give equal ranges.

Newton's second law

The vector sum of forces on a body equals its mass times acceleration, ΣF = ma, measured in an inertial frame. Weight is the force mg directed vertically downward.

Work–energy theorem

The net work done on a body equals its change in kinetic energy, W_net = ½mv² − ½mv₀². Work is F·d·cos θ, where θ is the angle between force and displacement.

Conservation of momentum

In a collision, total momentum is conserved when external impulses are negligible. Elastic collisions also conserve kinetic energy; perfectly inelastic collisions leave the bodies moving together.

Centripetal acceleration

A body moving on a circle of radius r at speed v has acceleration v²/r directed toward the centre. Its angular velocity is ω = 2π/T, and a = ω²r.

Static equilibrium conditions

A rigid body stays at rest when the vector sum of forces is zero and the sum of torques about any chosen point is also zero. Torque is r·F·sin θ.

Resistors in parallel

For resistors connected in parallel, the reciprocals add: 1/R = 1/R₁ + 1/R₂. The equivalent resistance is smaller than the smallest individual resistor, and the voltage is common.

Kirchhoff's loop rule

Around any closed circuit loop, the algebraic sum of potential differences is zero. Traverse each element, counting emfs and resistive drops with consistent signs, to build equations.

Energy stored in a capacitor

A capacitor of capacitance C holding charge Q at voltage U stores energy E = ½CU² = Q²/(2C) = ½QU. Capacitance is measured in farads.

Faraday–Lenz law

An induced emf equals the negative rate of change of magnetic flux, e = −dΦ/dt. The minus sign states that the induced current opposes the change producing it.

Total internal reflection

Light travelling from a denser to a less dense medium is totally reflected when the incidence angle exceeds the critical angle, whose sine equals n₂/n₁. It requires n₁ greater than n₂.

Molarity and dilution

Molar concentration c = n/V in mol per litre. On dilution the amount of solute is unchanged, so c₁V₁ = c₂V₂ for the same substance before and after adding solvent.

Oxidation number rules

An uncombined element has oxidation number zero, oxygen is usually −2 and hydrogen +1. The sum over a neutral molecule is zero; over a polyatomic ion it equals the ion charge.

Limiting reactant

The reactant that produces the smallest amount of product, given the balanced equation and the available moles. It is fully consumed and fixes the theoretical yield of the reaction.