# Differential & Integral Calculus, Linear Algebra & Differential Equations > Single-variable differential and integral calculus (limits, L'Hôpital's rule, derivatives, Fundamental Theorem of Calculus, Taylor series, improper integrals), linear algebra (matrix operations, determinants, rank-nullity, eigenvalues and eigenvectors), and first-order ordinary differential equations. - **Canonical URL:** https://flashcards.world/flashcards/sets/c506cbe5-56fc-4b5a-a547-a783ed72005d/ - **Markdown URL:** https://flashcards.world/flashcards/sets/c506cbe5-56fc-4b5a-a547-a783ed72005d.md - **Level:** advanced - **Languages:** en → en - **Total Cards:** 32 ## Flashcards (32 Cards) | # | Front (Term / Prompt) | Back (Definition / Answer) | Example | Multiple-Choice Options | | :--- | :--- | :--- | :--- | :--- | | 1 | What is the characteristic equation of ay'' + by' + cy = 0 with real constants and a ≠ 0? | Substituting y = e^(rx) gives ar² + br + c = 0. | | | | 2 | When is a real n × n matrix diagonalizable over R? | Exactly when it has n linearly independent real eigenvectors, so A = PDP⁻¹ with real invertible P and real diagonal D. Having n distinct real eigenvalues is sufficient. A real symmetric matrix is also diagonalizable over R. | | | | 3 | What geometric quantities enter the washer and cylindrical-shell volume formulas? | Washers: V = π∫ₐᵇ(R(x)² - r(x)²) dx, with outer radius R ≥ inner radius r ≥ 0. Shells: V = 2π∫ₐᵇ ρ(x)h(x) dx, with radius ρ ≥ 0 and height h ≥ 0. Choose bounds and slices that cover the solid without double counting; r = 0 gives disks. | | | | 4 | What solves y' = ky with constant k and y(0) = y₀? | y(t) = y₀e^(kt). For y₀ > 0, k > 0 gives growth and k < 0 gives decay; k = 0 gives a constant. If y₀ = 0, the solution remains zero. | | | | 5 | What does the spectral theorem say for a real symmetric matrix A? | It has real eigenvalues and an orthonormal eigenvector basis. Thus A = QDQᵀ, where D is real diagonal and QᵀQ = I. | | | | 6 | What does the Intermediate Value Theorem say for f continuous on [a, b], with a < b? | Every value u between f(a) and f(b), including the endpoints, equals f(c) for some c ∈ [a, b]. If u is strictly between the endpoint values, such a c lies in (a, b). | | | | 7 | What makes vectors v₁, …, vₖ linearly independent? | The only scalars satisfying c₁v₁ + … + cₖvₖ = 0 are c₁ = … = cₖ = 0. | | | | 8 | What is the general solution of ay'' + by' + cy = 0 for a repeated real characteristic root r? | With real constant coefficients and a ≠ 0: y(x) = (C₁ + C₂x)e^(rx), where C₁ and C₂ are arbitrary constants. | | | | 9 | How do you solve y' = g(x)h(y) by separation without losing constant solutions? | Where h(y) ≠ 0, integrate dy/h(y) = g(x) dx, giving ∫dy/h(y) = ∫g(x) dx + C. Separately check every root h(y*) = 0: y = y* is an equilibrium solution lost by division. Check each result on its valid interval. | | | | 10 | How does the second derivative test classify a stationary point c? | If f is twice differentiable near c and f'(c) = 0, then f''(c) > 0 gives a strict local minimum and f''(c) < 0 gives a strict local maximum. If f''(c) = 0, this test is inconclusive. | | | | 11 | When can an m × k matrix A multiply a p × n matrix B, and what size is AB? | The inner dimensions must match: k = p. The product AB then has size m × n. | | | | 12 | What integrating factor solves y' + P(x)y = Q(x) when P and Q are continuous on an interval? | μ(x) = exp(∫P(x) dx). Then (μy)' = μQ, so y = μ⁻¹(∫μQ dx + C) on that interval. | | | | 13 | How does an antiderivative evaluate a definite integral? | If f is continuous on [a, b] and F' = f, then ∫ₐᵇ f(x) dx = F(b) - F(a). | | | | 14 | For A = [[a, b], [c, d]], what is det(A), and how does \|det(A)\| act on area? | det(A) = ad - bc. For the linear map on R², \|det(A)\| is the area scale factor; a zero determinant collapses area to zero. | | | | 15 | How is the dot product in Rⁿ computed, and when does it express a right angle? | u · v = Σᵢuᵢvᵢ. For nonzero vectors it also equals \|\|u\|\|\|\|v\|\|cos θ, so u · v = 0 means θ = 90°. A zero vector has dot product zero with every vector, but no defined angle. | | | | 16 | For f: D → R and an accumulation point c of D, what does lim[x→c] f(x) = L mean? | For every ε > 0, there is δ > 0 such that every x ∈ D with 0 < \|x - c\| < δ satisfies \|f(x) - L\| < ε. | | | | 17 | What is the general solution of ay'' + by' + cy = 0 for two distinct real characteristic roots r₁, r₂? | With real constant coefficients and a ≠ 0: y(x) = C₁e^(r₁x) + C₂e^(r₂x), where C₁ and C₂ are arbitrary constants. | | | | 18 | What are the product and quotient differentiation rules? | For differentiable u and v: (uv)' = u'v + uv'. Where v ≠ 0, (u/v)' = (u'v - uv')/v². | | | | 19 | When is M(x, y) dx + N(x, y) dy = 0 exact, and how can this be tested? | Exactness means there is a potential F with Fₓ = M and Fᵧ = N; solutions satisfy F(x, y) = C. If M and N have continuous first partial derivatives on an open simply connected domain, exactness is equivalent to Mᵧ = Nₓ there. | | | | 20 | When does A = [[a, b], [c, d]] have an inverse, and what is it? | If ad - bc ≠ 0, then A⁻¹ = (1/(ad - bc))[[d, -b], [-c, a]]. If the determinant is zero, no inverse exists. | | | | 21 | What does rank–nullity state for A representing T: Rⁿ → Rᵐ? | rank(A) + nullity(A) = n. Rank is the dimension of the column space; nullity is the dimension of the kernel {x: Ax = 0}. | | | | 22 | What conditions allow L'Hôpital's rule for a quotient limit? | The quotient must have form 0/0 or ∞/∞; f and g must be differentiable on the relevant punctured interval, with g and g' nonzero there. If lim f'/g' exists or is ±∞, then lim f/g has that same value. One-sided and infinite-endpoint versions use the corresponding intervals. | | | | 23 | What is integration by parts, and which differentiation rule produces it? | ∫ u dv = uv - ∫ v du, with an arbitrary additive constant for an indefinite integral. It follows from the product rule. For definite integrals, use the boundary term [uv] at the limits. | | | | 24 | What is the chain rule for h(x) = f(g(x))? | If g is differentiable at x and f is differentiable at g(x), then h'(x) = f'(g(x))g'(x). | | | | 25 | What conditions express continuity at an interior point c of a real function's domain? | f(c) is defined, lim[x→c] f(x) exists, and that limit equals f(c). | | | | 26 | How is the derivative f'(x) defined at an interior point x? | f'(x) = lim[h→0] (f(x + h) - f(x))/h, provided this limit exists as a finite real number. | | | | 27 | What does the Fundamental Theorem of Calculus say about an accumulation integral? | If f is continuous on [a, b] and G(x) = ∫ₐˣ f(t) dt, then G is continuous on [a, b] and G'(x) = f(x) for a < x < b. | | | | 28 | What is a basis of a finite-dimensional vector space V, and what is dim(V)? | A basis is a linearly independent set that spans V. Every basis has the same number of vectors; that number is dim(V). | | | | 29 | How does the Wronskian identify a fundamental pair for y'' + p(x)y' + q(x)y = 0? | For two solutions on an interval where p and q are continuous, W = y₁y₂' - y₁'y₂. W ≠ 0 at one point is equivalent to their linear independence; they then form a fundamental pair, and every solution is C₁y₁ + C₂y₂ on that interval. | | | | 30 | How are an eigenvalue λ and an eigenvector v of a square matrix A defined and found? | They satisfy Av = λv with v ≠ 0. Eigenvalues solve det(A - λI) = 0; for each λ, eigenvectors are the nonzero solutions of (A - λI)v = 0. | | | | 31 | What is the real general solution of ay'' + by' + cy = 0 for roots α ± iβ with β > 0? | With real constant coefficients and a ≠ 0: y(x) = e^(αx)(C₁ cos(βx) + C₂ sin(βx)), where C₁ and C₂ are arbitrary real constants. | | | | 32 | State the Mean Value Theorem for a < b. | If f is continuous on [a, b] and differentiable on (a, b), some c ∈ (a, b) satisfies f'(c) = (f(b) - f(a))/(b - a). | | | --- Study this deck interactively with FSRS-6 spaced repetition, audio review, and multiple-choice mode at [Flashcards World](https://flashcards.world/flashcards/sets/c506cbe5-56fc-4b5a-a547-a783ed72005d/).