Maharashtra HSC Mathematics & Statistics - Arts & Science (MSBSHSE) flashcards

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

First principle derivative

The derivative of f at x is the limit as h approaches 0 of [f(x+h) − f(x)]/h, provided this limit exists and is finite.

Chain rule

If y depends on u and u depends on x, then dy/dx equals dy/du multiplied by du/dx. Used for composite functions and parametric curves.

Derivative of tan inverse x

d/dx of tan⁻¹x equals 1/(1+x²). The corresponding integral of 1/(1+x²) dx is tan⁻¹x + c, a standard result.

Logarithmic differentiation

Take the natural log of both sides of y = f(x), differentiate implicitly, then multiply the result by y to recover dy/dx. Best for products, quotients and variable powers.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a) equals f(b), then at least one c in (a,b) satisfies f'(c) = 0.

Mean value theorem

For f continuous on [a,b] and differentiable on (a,b), some c in (a,b) gives f'(c) equal to [f(b) − f(a)]/(b − a), the average slope.

Second derivative test

At a stationary point x = c, f''(c) less than 0 means local maximum, f''(c) greater than 0 means local minimum, and f''(c) = 0 is inconclusive.

Point of inflection

A point where the concavity of the curve changes sign, so f'' changes sign there. It is not necessarily a stationary point.

Integration by parts

Integral of u v dx equals u times integral of v dx minus integral of (u' times integral of v dx) dx. Choose u by the ILATE priority order.

Even and odd definite integrals

Over symmetric limits −a to a, the integral of an even function is twice the integral from 0 to a; for an odd function the integral is zero.

Area between two curves

If f(x) is at least g(x) on [a,b], the enclosed area equals the definite integral from a to b of [f(x) − g(x)] dx.

Volume of revolution

Rotating y = f(x) about the x-axis from a to b gives volume equal to pi times the integral from a to b of y squared dx.

Order and degree

Order is the highest derivative appearing in a differential equation. Degree is the power of that highest derivative after clearing radicals and fractions in derivatives.

Variable separable equation

An equation rearrangeable as f(y)dy = g(x)dx. Integrate both sides and add one arbitrary constant to obtain the general solution.

Integrating factor for linear ODE

For dy/dx + Py = Q, the integrating factor is e raised to the integral of P dx. The solution is y times this factor equals the integral of Q times the factor plus c.

Homogeneous differential equation

An equation where dy/dx can be written as a function of y/x alone. Substitute y = vx, giving dy/dx = v + x dv/dx, then separate variables.

Dot product

For a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, a·b equals a₁b₁ + a₂b₂ + a₃b₃, which also equals |a||b| cos θ.

Cross product

A vector perpendicular to both a and b with magnitude |a||b| sin θ. Computed as the determinant with i, j, k in the first row.

Scalar triple product

a·(b × c), written [a b c], equals the volume of the parallelepiped with edges a, b, c. It is zero when the three vectors are coplanar.

Distance from point to plane

For plane ax + by + cz + d = 0 and point (x₁,y₁,z₁), the perpendicular distance is |ax₁ + by₁ + cz₁ + d| divided by the square root of a² + b² + c².

Corner point method

In graphical linear programming, evaluate the objective function at every vertex of the feasible region. The optimum of a bounded region occurs at one of them.

Feasible region

The set of all points satisfying every constraint and the non-negativity conditions simultaneously. It is the shaded intersection of the half-planes.

Unbounded feasible region

A feasible region extending infinitely. A vertex optimum is accepted only if the objective line shares no point with the region beyond that value.

Contrapositive equivalence

The statement p implies q is logically equivalent to not-q implies not-p. The converse, q implies p, is not equivalent to the original.

Tautology

A compound statement that is true for every possible combination of truth values of its components. Its negation is a contradiction.

Inverse of a square matrix

A square matrix A has inverse adj(A) divided by determinant of A, existing only when the determinant is nonzero. For singular matrices no inverse exists.

Skew-symmetric matrix

A square matrix whose transpose equals its negative. Every diagonal entry must be zero, since each equals its own negative.

Binomial probability formula

In n independent trials with success probability p, P(X = r) equals C(n,r) p^r (1−p)^(n−r) for r from 0 to n.

Binomial mean and variance

For B(n,p), the mean is np and the variance is np(1−p). The standard deviation is the square root of that variance.

Coefficient of variation

Standard deviation divided by mean, multiplied by 100 and expressed as a percentage. It compares variability between data sets with different units.

Pearson's skewness coefficient

Mean minus mode, divided by standard deviation. When the mode is unreliable, mean minus median divided by standard deviation is used instead.

Correlation coefficient r

A number between −1 and 1 measuring strength and direction of linear association. Zero indicates no linear correlation, not necessarily independence.