AP ECET flashcards

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

Derivative of sin x

d/dx (sin x) = cos x, while d/dx (cos x) = −sin x. These are standard trigonometric derivatives, and the angle must be measured in radians.

Discriminant of a quadratic

For ax² + bx + c = 0 the discriminant is D = b² − 4ac. D > 0 gives two real distinct roots, D = 0 two equal real roots, and D < 0 a conjugate complex pair.

Integration by parts

∫u dv = uv − ∫v du. It is applied when the integrand is a product; choose u following the LIATE order: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential.

Perpendicular vector condition

Two non-zero vectors a and b are perpendicular exactly when their dot product a·b = |a||b|cos θ equals zero, that is when the angle between them is 90°.

Determinant of a 2x2 matrix

For A = [[a, b], [c, d]], det A = ad − bc. The matrix is invertible only when this value is non-zero; if det A = 0 the matrix is singular.

Binomial general term

In the expansion of (a + b)ⁿ the term containing bʳ is T(r+1) = nCr·aⁿ⁻ʳ·bʳ where nCr = n!/[r!(n−r)!]; the expansion has n + 1 terms.

Eccentricity of an ellipse

For x²/a² + y²/b² = 1 with a > b, the eccentricity is e = √(1 − b²/a²), which always lies between 0 and 1; the foci are at (±ae, 0).

Conditional probability formula

P(A|B) = P(A∩B)/P(B), defined when P(B) > 0. It gives the probability of A given that B has occurred and rearranges to the multiplication rule P(A∩B) = P(A|B)P(B).

Sum of first n naturals

1 + 2 + … + n = n(n+1)/2. The related results are Σn² = n(n+1)(2n+1)/6 and Σn³ = [n(n+1)/2]², used in series summation questions.

Newton's second law

The net external force on a body equals the rate of change of its momentum, F = dp/dt. For constant mass this reduces to F = ma, with force measured in newtons.

Escape velocity from Earth

v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s. It is the minimum launch speed needed to leave Earth's gravitational field without further propulsion and is independent of the projectile's mass.

Period of simple pendulum

T = 2π√(L/g) for small angular amplitude, where L is the effective length and g the local acceleration due to gravity. The period does not depend on the mass of the bob.

Carnot engine efficiency

η = 1 − T_c/T_h, where T_h and T_c are the source and sink temperatures in kelvin. No heat engine operating between these two reservoirs can exceed this efficiency.

Lens formula

1/f = 1/v − 1/u using Cartesian sign conventions, where f is focal length, v image distance and u object distance. A converging lens has a positive focal length.

Snell's law of refraction

n₁ sin θ₁ = n₂ sin θ₂, where n is the refractive index of each medium and the angles are measured from the normal. Light bends toward the normal on entering a denser medium.

Resistors in parallel

1/R_p = 1/R₁ + 1/R₂ + …, so the equivalent resistance is smaller than the smallest branch resistance, and the same potential difference appears across every branch.

First law of thermodynamics

ΔQ = ΔU + ΔW: heat supplied raises internal energy and/or does work by the system. In an isothermal process ΔU = 0; in an adiabatic process ΔQ = 0.

Avogadro number

One mole of any substance contains 6.022 × 10²³ elementary entities such as atoms, molecules or ions. It links the mass of a sample in grams to the number of particles present.

Molar volume at STP

One mole of an ideal gas occupies 22.4 litres at 273.15 K and 1 atm pressure. At 273.15 K and 1 bar the corresponding value is 22.7 litres.

Definition of pH

pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol/L. A pH below 7 at 25 °C indicates an acidic solution and a pH above 7 a basic one.

Le Chatelier's principle

When a system at equilibrium is disturbed by a change in concentration, pressure or temperature, the equilibrium shifts in the direction that counteracts the change; a catalyst does not shift it.

sp3 hybridization geometry

Four electron domains around a central atom give sp³ hybridization, a tetrahedral arrangement and bond angles of about 109.5°, as in methane; lone pairs compress the angle slightly.

Nernst equation at 25 C

E = E° − (0.0591/n) log Q, where n is the number of electrons transferred and Q the reaction quotient. It gives the cell potential when concentrations differ from standard conditions.

Hess's law

The enthalpy change of an overall reaction is the same whatever route is taken, so the ΔH values of individual steps can be added. This follows from enthalpy being a state function.

Slump test measures

The slump test measures the workability, or consistency, of fresh concrete. A standard cone is filled in layers, lifted vertically, and the settlement of the concrete is recorded in millimetres.

Maximum bending moment, simply supported UDL

For a simply supported beam carrying a uniformly distributed load w per unit length over span L, the maximum bending moment is wL²/8 at midspan and maximum shear force is wL/2 at the supports.

Bending stress formula

σ = My/I, where M is the bending moment, y the distance from the neutral axis and I the second moment of area. Maximum stress occurs at the extreme fibre where y = y_max.

Height of instrument in levelling

HI = reduced level of the station + backsight reading. The reduced level of the next station is then found by subtracting the foresight reading from the HI.

Void ratio of soil

e = V_v/V_s, the volume of voids divided by the volume of solids. It is usually expressed as a decimal and relates to porosity by n = e/(1 + e).

Porosity from void ratio

n = e/(1 + e), and conversely e = n/(1 − n), where e is the void ratio and n the porosity. Porosity is normally reported as a percentage.

Manning's equation

V = (1/n) R^(2/3) S^(1/2), where n is Manning's roughness coefficient, R the hydraulic radius equal to area divided by wetted perimeter, and S the bed slope.

Nominal mix for M20 concrete

The nominal volume ratio for M20 grade concrete is 1 : 1.5 : 3 (cement : fine aggregate : coarse aggregate), giving a characteristic cube strength of 20 N/mm² at 28 days.