Maharashtra HSC Mathematics & Statistics - Arts & Science (MSBSHSE) key terms

The limit of the difference quotient [f(x+h) − f(x)]/h as h tends to zero, giving the instantaneous rate of change of a function at a point.

The property that a function has a finite derivative at a point; it requires continuity there, but continuity alone does not guarantee it, as with |x| at zero.

A rule for differentiating a composite function: the derivative of f(g(x)) equals f′(g(x)) multiplied by g′(x), applied outward from the inner function.

A technique for finding dy/dx when a relation between x and y is not solved for y: differentiate both sides with respect to x, treating y as a function of x.

A method that takes the natural logarithm of both sides before differentiating, useful for products, quotients, and functions of the form [f(x)]^g(x).

A derivative obtained by differentiating repeatedly, such as the second derivative d²y/dx², which measures the rate of change of the first derivative.

If a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then some c in (a, b) satisfies f′(c) = 0.

If a function is continuous on [a, b] and differentiable on (a, b), then some c in (a, b) satisfies f′(c) = [f(b) − f(a)]/(b − a).

An interior point of the domain where the derivative is zero or does not exist; candidates for local maxima, local minima, or points of inflection.

At a critical point where f′(c) = 0, a negative second derivative gives a local maximum and a positive one gives a local minimum; zero is inconclusive.

A point where the curve changes concavity, so the second derivative changes sign; it need not be a stationary point.

The way a curve bends: concave upward where the second derivative is positive, concave downward where it is negative, over the relevant interval.

A differentiable function increases on an interval where its derivative is positive and decreases where its derivative is negative, tested between critical points.

The derivative of the total cost function with respect to output; it approximates the extra cost of producing one more unit.

The derivative of total revenue with respect to quantity sold; profit is maximised where marginal revenue equals marginal cost.

The family of all antiderivatives of a function, written with an arbitrary constant of integration, since differentiating any member returns the integrand.

The limit of a Riemann sum over an interval [a, b]; it gives a signed area and is evaluated using an antiderivative and the fundamental theorem of calculus.

A rule derived from the product rule: the integral of u dv equals uv minus the integral of v du, chosen so the remaining integral is simpler.

The region between a curve and the x-axis over an interval, computed by a definite integral; portions below the axis contribute negative signed area.

The region enclosed by two curves over an interval, found by integrating the upper curve minus the lower curve between their intersection points.

The order of the highest derivative appearing in the equation; a second-order equation contains d²y/dx² but no higher derivative.

The power of the highest-order derivative once the equation is written as a polynomial in derivatives with rational exponents cleared.

The solution of a differential equation containing as many arbitrary constants as the order of the equation, representing a whole family of curves.

A solution obtained from the general solution by fixing the arbitrary constants using given initial or boundary conditions.

A first-order differential equation that can be rearranged so all y-terms sit with dy and all x-terms with dx, then integrated on both sides.

A first-order equation of the form dy/dx + Py = Q, with P and Q functions of x, solved using the integrating factor e^∫P dx.

The dot product of one vector with the cross product of two others; its absolute value equals the volume of the parallelepiped they span.

The cross product of two vectors, giving a vector perpendicular to both whose magnitude equals the area of the parallelogram they span.

The cosines of the angles a line or vector makes with the positive x, y, and z axes; their squares sum to one.

Any three numbers proportional to the direction cosines of a line; they fix the line's direction but are not unique.

Three vectors lie in one plane exactly when their scalar triple product is zero, equivalently when they are linearly dependent.

The length of the common perpendicular between two non-intersecting, non-parallel lines, computed from a vector and scalar triple product formula.

The set of points satisfying all constraints of a linear programming problem simultaneously, usually a convex polygon in two variables.

A procedure for solving a linear programming problem by evaluating the objective function at every vertex of the feasible region.

The linear expression in the decision variables that is to be maximised or minimised, subject to the stated constraints.

A compound statement that is true for every possible truth-value assignment to its component statements; its negation is a contradiction.

Two statements are logically equivalent when they have identical truth values in every row of their joint truth table.

For an implication p implies q, the converse is q implies p and the contrapositive is not-q implies not-p; only the contrapositive is equivalent to the original.

The adjoint is the transpose of the cofactor matrix; for a square matrix with nonzero determinant, the inverse equals the adjoint divided by the determinant.

A scalar assigned to a square matrix that indicates whether the matrix is invertible and measures how it scales areas or volumes.

A square matrix whose determinant is zero; it has no inverse, and its rows or columns are linearly dependent.

A real-valued function defined on the sample space of an experiment, assigning a number to each outcome.

A function giving the probability that a discrete random variable equals each of its possible values; the probabilities are non-negative and sum to one.

The probability-weighted average of a random variable's values, written E(X); for a discrete variable it is the sum of x times P(X = x).

The expected value of the squared deviation from the mean, E(X²) − [E(X)]², measuring how spread out the distribution is.

The distribution of the number of successes in n independent Bernoulli trials with constant success probability p, with parameters n and p.

A single experiment with exactly two mutually exclusive outcomes, conventionally called success and failure, with fixed probabilities.

The positive square root of the variance, expressed in the same units as the data and widely used as the main measure of spread.

The ratio of standard deviation to mean, often given as a percentage; it compares variability between data sets with different units or means.

A measure of asymmetry equal to (mean − mode) divided by standard deviation, or equivalently three times (mean − median) divided by standard deviation.

A quartile-based measure of asymmetry, (Q3 + Q1 − 2Q2)/(Q3 − Q1), useful when the mode is ill-defined or the data are open-ended.

A number between minus one and plus one measuring the strength and direction of the linear relationship between two variables, independent of units.

Two best-fit lines, of y on x and x on y, found by least squares; they intersect at the means of both variables.