TS ECET flashcards

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

Limit of sin x over x

As x approaches zero, sin x divided by x approaches 1. This standard limit underlies the derivative of sine and is evaluated in radians, not degrees.

L'Hôpital's rule condition

For a 0/0 or ∞/∞ indeterminate form, the limit of f/g equals the limit of f'/g' provided the derivatives exist and the resulting limit exists.

Rolle's theorem

If a function is continuous on a closed interval, differentiable inside it, and equal at both endpoints, then at least one interior point has zero derivative.

Integration by parts formula

The integral of u dv equals uv minus the integral of v du. Choose u following the LIATE order to make the remaining integral simpler.

Eigenvalue trace relation

For any square matrix, the sum of eigenvalues equals the trace and the product equals the determinant. This gives a quick check on computed roots.

Scalar triple product meaning

The dot product of vector a with the cross product of b and c gives the signed volume of the parallelepiped formed by the three vectors.

Projectile time of flight

For launch speed u at angle θ on level ground, time of flight is 2u sinθ/g and horizontal range is u² sin2θ/g, with maximum range at 45 degrees.

Simple pendulum period

A simple pendulum of length L in gravity g has period 2π times the square root of L/g. The period is independent of bob mass.

Moment of inertia of disc

A uniform solid disc or cylinder rotating about its central axis has moment of inertia one-half M R squared, where M is mass and R is radius.

Kirchhoff loop rule

Around any closed circuit loop, the algebraic sum of EMFs equals the sum of potential drops from current through resistances, expressing energy conservation.

Photoelectric threshold frequency

The minimum light frequency that ejects electrons from a metal surface. Below it no emission occurs regardless of intensity; it equals work function divided by Planck's constant.

de Broglie wavelength

A particle of momentum p has wavelength h/p, where h is Planck's constant. Electrons accelerated through V volts show about 12.27/√V angstroms.

Radioactive half-life

The time for half the nuclei in a sample to decay, equal to ln2 divided by the decay constant. Activity also halves each half-life.

Lens magnification sign

In the Cartesian convention, magnification is image distance divided by object distance. A negative value means the image is real and inverted.

Ideal gas molar volume

At standard temperature and pressure, one mole of an ideal gas occupies about 22.4 litres, following from PV = nRT with R = 8.314 J/mol·K.

Molarity dilution relation

For dilution of a solution, initial molarity times initial volume equals final molarity times final volume, because the moles of solute stay constant.

Buffer pH equation

Henderson–Hasselbalch gives pH equal to pKa plus the log of salt concentration over acid concentration for an acidic buffer system.

Bond order from MO theory

Half the difference between bonding and antibonding electron counts. Oxygen has bond order two and is paramagnetic because of unpaired pi-star electrons.

Hybridisation steric number

Steric number counts sigma bonds plus lone pairs on the central atom: 2 gives sp, 3 gives sp², 4 gives sp³, and 5 gives sp³d geometry.

Markovnikov's rule

When hydrogen halide adds to an unsymmetrical alkene, hydrogen attaches to the carbon bearing more hydrogens, producing the more stable carbocation intermediate.

SN1 versus SN2 mechanism

SN1 is a two-step carbocation pathway whose rate depends only on substrate and is favoured by tertiary halides. SN2 is a single backside displacement favoured by primary halides.

Faraday's law of electrolysis

Mass deposited equals molar mass times current times time, divided by n times Faraday's constant, where n is electrons per ion.

Thevenin equivalent circuit

Any linear two-terminal network reduces to one voltage source in series with one resistance, found from open-circuit voltage and deactivated-source resistance.

Transformer EMF equation

Induced EMF equals 4.44 times frequency times turns times peak flux. Voltage ratio equals turns ratio; current ratio is its inverse.

Three-phase power formula

Balanced three-phase real power is √3 times line voltage times line current times power factor, valid for both star and delta connected loads.

BJT active region bias

A bipolar transistor operates in forward-active mode when the base–emitter junction is forward biased and the base–collector junction is reverse biased.

Op-amp virtual short rule

With negative feedback and ideal infinite gain, the two input terminals sit at nearly equal voltage and draw no current, simplifying gain calculations.

Inverting amplifier gain

An ideal op-amp inverting configuration has voltage gain equal to negative feedback resistance divided by input resistance, with the output phase inverted.

Binary search complexity

Searching a sorted array by repeatedly halving the interval takes O(log n) comparisons, far fewer than linear search for large arrays.

String terminator in C

Every C string literal ends with an implicit null character, so storing n visible characters requires an array of n+1 elements.

Isothermal ideal-gas work

For a reversible isothermal expansion of an ideal gas, work done is nRT times the natural log of final volume over initial volume.

First law of thermodynamics

Heat added to a system equals the change in internal energy plus work done by the system, a statement of energy conservation for thermodynamic processes.