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Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.

Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.

For the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ to be valid, the function $f(x)$ must be continuous on the closed interval $[a, b]$, and its derivative $f'(x)$ must also be continuous on $[a, b]$. These conditions ensure that the curve is smooth enough for the integral to accurately represent its length.

Conditions: The curve is defined by a function $y = f(x)$.; The interval of integration is $[a, b]$.; $f(x)$ is continuous on $[a, b]$.; $f'(x)$ is continuous on $[a, b]$.

Geometrically, the determinant of $a 2\times 2$ matrix represents the factor by which the linear transformation scales areas. Specifically, it is the signed area of the parallelogram formed by the matrix's column vectors.

Conditions: The matrix is $2\times 2$.; The transformation is linear.; Use ordinary Euclidean area in standard orthonormal coordinates.

Matrix-vector multiplication is reframed as scaling the transformed basis vectors (which are the columns of the matrix) by the input vector's components and summing them. This constructs the final position within the skewed coordinate system defined by the matrix, rather than just following a tedious arithmetic recipe.

Conditions: Input and output coordinates use the fixed standard basis.; The matrix represents an active linear map moving vectors.

To find the left-hand limit as x approaches a value c (denoted $\lim_{x \to c^-} f(x)$), observe the behavior of the function's curve strictly for input values less than c, moving towards c. If the y-values grow without bound in either the positive or negative direction, the finite real limit does not exist.

Conditions: The function has a vertical asymptote or discontinuity at $x=c$.; You are evaluating the limit from the left side (inputs strictly less than c).

To find the x-intercepts of the curve $y = x^2 - 4$, set the function equal to zero ($x^2 - 4 = 0$) and factor the expression as a difference of squares into $(x+2)(x-2) = 0$. Applying the zero-product property yields the solutions $x = -2$ and $x = 2$, which are the points where the graph crosses the x-axis.

Conditions: The curve is defined by the quadratic equation $y = x^2 - 4$.; The goal is to find where the graph crosses the x-axis.; The algebraic method used is factoring the difference of perfect squares.

For a differentiable function $f(x,y)$, the directional derivative along the vector $v=[-1,2]$ is calculated by multiplying the partial derivative with respect to $x$ by $-1$ and the partial derivative with respect to $y$ by $2$, then summing these products. This yields the expression $-\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$, assuming the partial derivatives exist.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $v=[-1,2]$.; The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

Applying matrix B and then matrix A composes the linear transformations. Since each transformation multiplies the area/volume by its respective determinant (including sign/orientation), the total scaling factor is the product $\det(A)\det(B)$.

Conditions: Matrices A and B are square and compatible for multiplication; Determinants are defined

In 3D, the determinant measures the volume scaling factor of a unit cube mapped to a parallelepiped. The sign indicates orientation: positive preserves the right-handed frame, while negative reverses it.

Conditions: Linear transformation in 3D space; Unit cube input

To find the point on the curve $y = x^2 - 4$ at $x = -1$, substitute $x = -1$ into the equation. Evaluating $(-1)^2 - 4$ yields $1 - 4 = -3$, so the coordinates of the point are $(-1, -3)$.

Conditions: The curve is defined by the equation $y = x^2 - 4$.; The x-coordinate of the desired point is given as $-1$.; The point must lie on the curve.

A set of vectors is linearly dependent if one of the vectors is redundant, meaning it provides no new directional freedom. Mathematically, this occurs when a vector lies perfectly flat on the plane (or line) already created by the others, allowing it to be written as a combination of the others.

Conditions: The set contains at least two vectors.; One vector lies in the span of the other vectors.

The limit definition of a derivative is the finite real limit of the difference quotient as the nonzero increment tends to zero. At an interior point where this limit exists, it represents the derivative and the tangent slope.

Conditions: A finite real derivative limit must exist at the point.; The point is interior and $Δx\ne 0$ while taking the limit.