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.

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.

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.

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.

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.

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).

What makes vectors v₁, …, vₖ linearly independent?

The only scalars satisfying c₁v₁ + … + cₖvₖ = 0 are c₁ = … = cₖ = 0.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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}.

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.

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.

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).

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).

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.

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.

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).

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.

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.

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.

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).

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.

Level: advanced
Languages: en → en
Cards: 32
Markdown: Differential & Integral Calculus, Linear Algebra & Differential Equations (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
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).