Balearic Islands PAU Mathematics II flashcards

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

Regular matrix

A square matrix whose determinant is non-zero. Only regular matrices have an inverse, and the associated homogeneous system has only the trivial solution.

Determinant of a 2x2 matrix

For the matrix with rows (a, b) and (c, d), the determinant equals ad minus bc. The matrix is invertible exactly when this value is not zero.

Rank by minors

The rank of a matrix is the order of its largest non-zero minor. It equals the number of non-zero rows left after Gaussian elimination and bounds the number of independent equations.

Rouché–Capelli theorem

A linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix. If that common rank is below the number of unknowns, the system has infinitely many solutions.

Cramer's rule

For a square system with non-zero determinant, each unknown equals the determinant of the matrix with its column replaced by the constants, divided by the original determinant.

Inverse of a 2x2 matrix

Swap the diagonal entries, change the sign of the off-diagonal entries, and divide by the determinant. This requires the determinant to be non-zero.

Scalar triple product

The dot product of one vector with the cross product of the other two. Its absolute value is the volume of the parallelepiped, and it is zero exactly when the three vectors are coplanar.

Angle between two lines

The acute angle whose cosine equals the absolute value of the dot product of the direction vectors divided by the product of their moduli.

Distance from point to plane

Substitute the point into the general plane equation, take the absolute value, and divide by the modulus of the normal vector given by the coefficients.

Skew lines

Two lines in space that are neither parallel nor intersecting, so they lie in no common plane. Their distance equals the triple product divided by the modulus of the cross product of directions.

Relative position of two planes

Compare their normal vectors. Proportional normals with proportional independent terms give coincident planes; proportional normals with different independent terms give parallel planes; otherwise the planes meet in a line.

Symmetric point about a plane

Project the point onto the plane along the normal direction, then reflect it: twice the projection minus the original point. The segment joining the two points is perpendicular to the plane.

Equation of a plane

A plane is determined by a point and a normal vector, giving A(x minus x0) plus B(y minus y0) plus C(z minus z0) equals zero, where (A, B, C) is the normal.

L'Hôpital's rule

For indeterminate quotients of the form zero over zero or infinity over infinity, the limit equals the limit of the quotient of the derivatives, provided that this second limit exists.

Differentiability implies continuity

If a function has a derivative at a point then it is continuous there. The converse fails: the absolute value function is continuous at zero but has no derivative there.

Chain rule

The derivative of a composite function equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Second derivative test

At a stationary point, a positive second derivative gives a local minimum and a negative one gives a local maximum. A zero second derivative is inconclusive.

Inflection point

A point where the concavity of the function changes sign. It requires the second derivative to vanish or fail to exist and to change sign across that point.

Global extrema on closed interval

Compare the values of the function at all critical points inside the interval and at both endpoints. The largest value is the global maximum and the smallest is the global minimum.

Oblique asymptote

A line y equals mx plus n approached at infinity, where m is the limit of f(x) over x and n is the limit of f(x) minus mx. Both limits must be finite and m non-zero.

Rolle's theorem

If a function is continuous on a closed interval, differentiable inside it, and takes equal values at the endpoints, then its derivative vanishes at some interior point.

Fundamental theorem of calculus

The definite integral of a continuous function over an interval equals the difference of any antiderivative evaluated at the upper and lower limits.

Integration by parts

The integral of u dv equals uv minus the integral of v du. It is used for products such as a power times an exponential, a logarithm, or a trigonometric function.

Area between two curves

Integrate the absolute value of the difference of the functions between their intersection abscissas. Split the integral wherever the curves cross inside the interval.

Volume by discs

Rotating the region under y equals f(x) about the x-axis gives volume equal to pi times the integral of the square of f between the limits.

Volume by cylindrical shells

Rotating about the y-axis, the volume equals two pi times the integral of x times f(x) between the limits, where x is the distance to the axis.

Change of limits in substitution

When substituting in a definite integral, transform the limits to the new variable instead of returning to the original variable, so the result is obtained directly.

Improper integral convergence

An integral over an unbounded interval is defined as a limit. The integral of one over x to the p from one to infinity converges exactly when p exceeds one.

Binomial distribution mean and variance

For n independent trials with success probability p, the mean is np and the variance is np(1 minus p). The standard deviation is the square root of the variance.

Bayes' theorem

The conditional probability of a cause given an observed event equals the prior probability of the cause times the likelihood, divided by the total probability of the event.

Normal approximation to binomial

When np and n(1 minus p) both exceed five, the binomial count is approximately normal with mean np and standard deviation the square root of np(1 minus p), using continuity correction.

Standard error of the mean

The standard deviation of the sample mean equals the population standard deviation divided by the square root of the sample size. It measures the precision of the estimate.