TS ECET key terms

The value a function approaches as its input approaches a specified point, regardless of the value at that point itself. It underpins continuity, derivatives, and integrals in calculus.

A property of a function whose value at a point equals the limit of the function there, with no break, jump, or hole. Checking it requires the limit to exist from both sides and match the function value.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of the difference quotient. Geometrically it gives the slope of the tangent line at a point.

The condition that a function has a well-defined derivative at a point, meaning its graph has a non-vertical tangent there. Differentiability at a point implies continuity there, but not the reverse.

The limit of a sum of function values times small interval widths over a range, interpreted as signed area under a curve. It is evaluated using antiderivatives via the fundamental theorem of calculus.

A function whose derivative equals a given function. Every continuous function has infinitely many antiderivatives differing by an additive constant, forming the indefinite integral.

A technique for integrating products, derived from the product rule, expressed as the integral of u dv equals uv minus the integral of v du. It works best when one factor simplifies on differentiation.

A scalar computed from a square matrix that indicates whether the matrix is invertible and scales volumes under the associated linear transformation. A zero determinant means the matrix is singular.

The matrix that, multiplied by the original square matrix, yields the identity matrix. It exists only when the determinant is nonzero and is computed using the adjugate divided by the determinant.

The number of linearly independent rows or columns in a matrix, equal to the order of the largest nonzero minor. It determines whether a system of linear equations has unique, infinite, or no solutions.

A scalar for which a nonzero vector, when multiplied by a square matrix, is merely scaled by that scalar. Found by solving the characteristic equation obtained from the determinant of the matrix minus lambda times identity.

A theorem stating that for a function continuous on a closed interval and differentiable inside it, some interior point has a derivative equal to the average rate of change over the interval.

Scalars have magnitude only, such as mass or temperature, while vectors have both magnitude and direction, such as force or velocity. Vector addition follows the triangle and parallelogram laws.

A scalar obtained by multiplying corresponding components of two vectors and summing them, equal to the product of magnitudes times the cosine of the angle between them. It is zero for perpendicular vectors.

A vector perpendicular to two given vectors, with magnitude equal to the product of their magnitudes times the sine of the angle between them. It is zero for parallel vectors and follows the right-hand rule.

Three principles: a body stays at rest or in uniform motion unless acted on by a net force; force equals mass times acceleration; and every action has an equal and opposite reaction.

The principle that total momentum of an isolated system remains constant because internal forces occur in equal and opposite pairs. It applies to collisions and explosions in the absence of external net force.

The statement that net work done on an object equals its change in kinetic energy. It links force applied over a displacement to the resulting speed change of the body.

The principle that the sum of kinetic and potential energy stays constant in a system where only conservative forces act. It allows solving motion problems without tracking forces at every instant.

A contact force opposing relative motion or its tendency between surfaces. Static friction adjusts up to a maximum proportional to the normal force; kinetic friction is roughly constant during sliding.

Oscillation where restoring force is proportional to displacement and directed toward equilibrium, producing sinusoidal motion. Its period is independent of amplitude and depends on mass and stiffness.

A condition where a system driven at a frequency near its natural frequency oscillates with large amplitude. It arises because the driving force stays in phase with the system's motion.

A relation describing wave propagation, connecting speed to frequency and wavelength, or relating second derivatives in space and time. It applies to sound, light, and waves on strings.

The relation stating that current through a conductor is directly proportional to potential difference across it and inversely proportional to resistance, at constant temperature. It fails for non-ohmic devices like diodes.

Two circuit rules: the sum of currents entering a junction equals the sum leaving, and the sum of potential differences around any closed loop is zero. They follow from charge and energy conservation.

The rate at which electrical energy is transferred in a circuit, equal to voltage times current and also expressible as current squared times resistance. It is measured in watts.

The production of an electromotive force in a conductor when magnetic flux through it changes, described by Faraday's law and given direction by Lenz's law. It is the basis of generators and transformers.

Atomic number counts protons and identifies the element; mass number counts protons plus neutrons and identifies the isotope. Electrons equal protons in a neutral atom.

Atoms of the same element with equal proton counts but different neutron counts, so they share chemical behavior but differ in mass. Some are radioactive and decay at characteristic half-lives.

The arrangement of electrons among atomic orbitals, following the Aufbau principle, Pauli exclusion principle, and Hund's rule. It explains periodic trends and chemical bonding behavior.

A measure of how strongly an atom attracts shared electrons in a bond. Large differences between bonded atoms produce ionic character; small differences produce nonpolar covalent bonds.

The mixing of atomic orbitals to form equivalent new orbitals that explain observed molecular geometry, such as sp, sp2, and sp3 arrangements. It predicts bond angles and shapes.

The existence of compounds with the same molecular formula but different structures or spatial arrangements. Structural isomers differ in connectivity; stereoisomers differ only in three-dimensional placement.

A counting unit for particles equal to Avogadro's number, linking mass, molar mass, and number of particles. It converts between grams of substance and numbers of atoms or molecules.

Concentration expressed as moles of solute per liter of solution. It is temperature-dependent because solution volume changes with temperature, unlike molality.

Calculation of quantitative relationships between reactants and products using balanced chemical equations and mole ratios. It predicts limiting reagents and theoretical product yields.

A logarithmic measure of hydrogen ion concentration, where pH equals the negative logarithm of that concentration. Values below seven are acidic, seven neutral, and above seven basic at room temperature.

A solution that resists pH change when small amounts of acid or base are added, typically containing a weak acid with its conjugate base. It works best near the acid's pKa value.

The time required for half the atoms in a radioactive sample to decay, or for a reactant concentration to fall to half its initial value in a first-order process. It is constant for first-order kinetics.