What would you like to understand?

Subtraction is equivalent to adding the additive inverse. In matrix algebra, multiplying a matrix by the scalar -1 creates its additive inverse.

Conditions: Scalar multiplication by -1 is defined entrywise.; Matrix addition is defined entrywise.; A and B have the same dimensions.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.

The formal definition states that the limit of $f(x)$ as $x$ approaches $a$ is $L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$. This rigorously captures the intuitive idea that $f(x)$ can be made arbitrarily close to $L$ by choosing $x$ sufficiently close to $a$ (but not equal to $a$).

Conditions: $\epsilon$ is an arbitrary positive real number; $\delta$ is a positive real number dependent on $\epsilon$; $x$ is in the domain of $f$ and $x \neq a$

When evaluating the limit of a partial sum $S_n$ expressed as a quotient of polynomials in $n$ as $n \to \infty$, dividing both the numerator and denominator by the highest power of $n$ in the denominator (specifically $n^2$) isolates the asymptotic behavior of each term. This transformation converts lower-order terms into fractions with $n$ in the denominator, which clearly tend to 0, thereby revealing that the numerator dominates and the limit is infinity.

Conditions: The expression is a quotient of polynomials in $n$.; The limit is taken as $n \to \infty$.; The highest power of $n$ in the denominator is $n^2$.

The notation $\frac{d}{dx}f(-2)$ represents evaluating the derivative of the function $f$ specifically at the horizontal input coordinate $x = -2$. It denotes the instantaneous slope of the tangent line to the graph at that exact abscissa, rather than referring to a point where the output or function value happens to be $-2$.

Conditions: The expression evaluates the derivative operator applied to the function $f$.; The argument inside the parentheses is strictly the independent variable (input $x$).

The real domain of the function $y=\sqrt{9-x^2}$ is the closed interval $[-3, 3]$. This restriction exists because, over the real numbers, the expression inside a square root (the radicand) must be nonnegative for the principal square root to be real-valued.

Conditions: Real-valued interpretation of the square root.; Using the principal square root convention.; The radicand is $9 - x^2$.

The exact value of the definite integral $\int_{-3}^{3} \sqrt{9-x^2}\,dx$ is $\frac{9\pi}{2}$. This is derived by interpreting the integral as the geometric area of the upper semicircle of a circle with radius 3.

Conditions: Interpret the integral as area under the graph on $[-3,3]$.; Recognize the graph as the upper semicircle of radius 3.; Exact equality for the displayed definite integral.

The infinitesimal base area $\Delta A_k = \Delta x_k \Delta y_k$ represents the footprint or base of a single representative rectangular box in the Riemann sum. It is the area of the $k$-th subrectangle in the domain partition.

Conditions: The domain is partitioned into subrectangles.; $\Delta x_k$ and $\Delta y_k$ are the side lengths of the $k$-th subrectangle.

Starting from $Av = \lambda v$, we rewrite the right side as $(\lambda I)v$ and move all terms to one side to get $(A - \lambda I)v = 0$. Since we seek non-zero solutions for $v$, the matrix $(A - \lambda I)$ must squash space into a lower dimension (have a non-trivial null space).

Conditions: $v$ is a non-zero eigenvector; $A$ is a square matrix; $I$ is the identity matrix

For n distinct objects, successive choices are n, $n-1$, …, 1, giving n!. The counting rule needs the stated number of choices after each preceding choice; this is not a probability-independence assumption.

Conditions: Objects are distinct.; Order matters.

The indefinite integral of $x^2$ yields a family of functions $F(x) = \frac{x^3}{3} + C$. By defining the accumulation function $A(x)$ such that $A(0) = \int_0^0 t^2 dt = 0$, we impose an initial condition.

Conditions: Integrand is $f(t) = t^2$; Lower limit of integration is 0; Accumulation starts from zero

The units of the area moment of inertia are derived directly from its defining integral, $I = \int r^2 dA$. The differential area element $dA$ has units of length squared (e.g., $m^2$).

Conditions: Lengths are measured in meters in the specific example.; The integral represents an area moment of inertia.