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The standard rule (smaller denominator implies larger fraction) applies strictly when the common numerator is positive. If the common numerator is zero, all such fractions are equal to zero, resulting in equality.

Conditions: Comparing fractions $\frac{n}{a}$ and $\frac{n}{b}$ with $0 < a < b$; Case 1: $n > 0$ (standard case); Case 2: $n = 0$; Case 3: $n < 0$

Acceleration is defined as the second derivative of the position (or displacement) function $s(t)$ with respect to time $t$. It represents the rate of change of velocity, where velocity itself is the first derivative of displacement.

Conditions: Motion occurs along a fixed line with a chosen coordinate direction.; The position function $s(t)$ is twice differentiable with respect to time.

The definition requires $x \neq a$ (expressed as $0 < |x - a|$) because the limit describes the behavior of the function *as it approaches* the point $a$, not its value *at* the point $a$. The function might be undefined at $x = a$, or its value $f(a)$ might differ from the limit $L$.

Conditions: The limit is concerned with the trend of $f(x)$ near $a$.; $f(a)$ may be undefined or discontinuous at $a$.

To find the equation of any line, you need two pieces of information: a point on the line (with an x, y coordinate) and the slope of the line. Once these are known, the equation can be constructed using forms like the point-slope formula.

Conditions: The object is a straight line in a 2D Cartesian coordinate system.; The goal is to determine its algebraic equation.

The residual vector $\vec{r} = A\vec{x}^* - \vec{b}$ is orthogonal to the column space $C(A)$ because $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$. Orthogonality to $C(A)$ means $\vec{r}$ is in the orthogonal complement $C(A)^\perp$.

Conditions: $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$; $C(A)^\perp = N(A^T)$ (Fundamental Theorem of Linear Algebra); Matrix multiplication distributes over subtraction

The determinant is negative because the transformation flips the relative ordering of the basis vectors (e.g., $\hat{i}$ rotating past $\hat{j}$), reversing the handedness of the coordinate system. The absolute value still gives the area scale, but the sign encodes this directional reversal.

Conditions: Transformation mirrors or reflects space; Basis vector order is inverted

The directional derivative generalizes the partial derivative by replacing axis-aligned displacements with an arbitrary direction vector scaled by a small scalar $h$. It measures the infinitesimal output change as $h$ approaches zero, extending the concept of partial derivatives from coordinate axes to any vector direction in the input plane.

Conditions: The function is a scalar-valued function of two variables.; The partial derivative is defined along coordinate axes.; The directional derivative uses an arbitrary vector direction.

The partial sum $S_n$ is the finite sum of the first $n$ terms, whereas the infinite series $S$ is defined as the limit of $S_n$ as $n \to \infty$. While $S_n$ forms a sequence of values, $S$ represents the single limiting value (or divergence) that this sequence approaches.

Conditions: $S_n$ denotes the sum of the first $n$ terms.; $S$ denotes the infinite series $\sum_{n=1}^{\infty} a_n$.; The limit is taken as $n \to \infty$.

Matrix addition requires matching dimensions so that every entry in one matrix has exactly one corresponding partner in the other matrix. Without identical row and column counts, some positions would lack a pair to add, making the operation undefined under standard rules.

Conditions: Standard definition of matrix addition applies.; Matrices represent rectangular arrays of numbers.

The Pythagorean theorem can be used because the perpendicular component displacements of a vector in an orthonormal Cartesian plane form the two legs of a right triangle, with the vector itself acting as the hypotenuse. The theorem relates the lengths of the legs to the length of the hypotenuse, allowing the calculation of the vector's magnitude (length).

Conditions: The coordinate axes are perpendicular (orthonormal).; The vector is drawn from the origin to its terminal point.; The magnitude is the Euclidean length of the vector.

The sample point $(x_k,y_k)$ serves as the specific input location where the surface function $f(x,y)$ is evaluated to determine the vertical height of the representative rectangular box. Geometrically, it anchors the top of the box to the surface $z=f(x,y)$ directly above that point, allowing the box's volume to be calculated as base area times this specific height.

Conditions: A subrectangle has been defined in the domain.; A point $(x_k,y_k)$ is chosen inside that subrectangle.

The greatest common factor (GCF) is the largest positive divisor shared by all of the compared positive integers. It is the maximum value in the set of their common factors.

Conditions: At least two positive integers are compared.; Positive divisors are compared.