State Part 2 of the Fundamental Theorem of Calculus (evaluating a definite integral).

If f is continuous on [a, b] and F is any antiderivative of f (F' = f), then ∫[a to b] f(x) dx = F(b) - F(a).

State Part 1 of the Fundamental Theorem of Calculus (differentiating an accumulator function).

If f is continuous on [a, b], then g(x) = ∫[a to x] f(t) dt is continuous on [a, b], differentiable on (a, b), and g'(x) = (d/dx) ∫[a to x] f(t) dt = f(x).

What is the Integration by Parts formula, and what differentiation rule is it derived from?

∫ u dv = u*v - ∫ v du; derived directly by integrating the product rule d(uv) = u*dv + v*du.

Under what conditions can L'Hôpital's Rule be applied to evaluate lim [x -> c] f(x) / g(x)?

When the quotient produces the indeterminate form 0/0 or ±∞/±∞ and f, g are differentiable near c; then lim f(x)/g(x) = lim f'(x)/g'(x) if the latter limit exists.

What is the general solution to ay'' + by' + cy = 0 when the characteristic equation has two distinct real roots r1 != r2?

y(x) = c1 * e^(r1*x) + c2 * e^(r2*x).

Define an eigenvalue λ and corresponding eigenvector v of an n x n square matrix A.

A non-zero vector v such that A*v = λ*v; found by solving the characteristic equation det(A - λ*I) = 0.

What does the Wronskian determinant W(y1, y2)(x) determine about two solutions to a linear second-order ODE?

W(y1, y2) = y1*y2' - y1'*y2; if W != 0 at any point, the solutions y1 and y2 are linearly independent and form a fundamental solution set.

What is the dot product of two vectors u and v in R^n, and what geometric property does u · v = 0 indicate?

u · v = Σ [u_i * v_i] = ||u|| * ||v|| * cos(θ); u · v = 0 indicates that the non-zero vectors are orthogonal (perpendicular, θ = 90°).

Define a basis of a vector space V and state what the dimension of V represents.

A basis is a linearly independent set of vectors that spans V; the dimension dim(V) is the unique number of vectors in any basis of V.

What does the Spectral Theorem state for real symmetric matrices (A = A^T)?

Every real symmetric matrix has all real eigenvalues and can be orthogonally diagonalized: A = Q * D * Q^T, where Q is an orthogonal matrix (Q^T = Q^-1).

What does the Intermediate Value Theorem (IVT) state for a continuous function on [a, b]?

If f is continuous on [a, b] and u is any value between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = u.

What is the integrating factor μ(x) used to solve the first-order linear ODE y' + P(x)*y = Q(x)?

μ(x) = exp(∫ P(x) dx); multiplying gives (μ*y)' = μ*Q, yielding y(x) = (1 / μ(x)) * [∫ μ(x)*Q(x) dx + C].

What is the general solution to the exponential growth/decay differential equation dy/dt = k*y with y(0) = y0?

y(t) = y0 * e^(k*t); represents exponential growth if k > 0 and exponential decay if k < 0.

What is the formula for the determinant of a 2x2 matrix A = [[a, b], [c, d]], and what does |det A| represent geometrically?

det(A) = ad - bc; |det(A)| represents the scaling factor by which linear transformation A multiplies areas in R^2.

What condition must be satisfied to multiply matrix A of dimension m x k by matrix B of dimension p x n?

The inner dimensions must be equal: k = p; the resulting matrix C = A*B has dimension m x n.

State the limit definition of the derivative f'(x) at a point x.

f'(x) = lim [h -> 0] (f(x + h) - f(x)) / h.

What is the characteristic equation for the second-order homogeneous linear ODE ay'' + by' + cy = 0 with constant coefficients?

a*r^2 + b*r + c = 0, obtained by testing solutions of the exponential form y = e^(r*x).

What does it mean for a set of vectors {v1, v2, ..., vk} to be linearly independent?

c1*v1 + c2*v2 + ... + ck*vk = 0 holds if and only if all scalar coefficients are zero: c1 = c2 = ... = ck = 0.

How is the volume of a solid of revolution computed using the disk method versus the cylindrical shell method?

Disk method (revolving around x-axis): V = π ∫ [R(x)]^2 dx; Shell method (revolving around y-axis): V = 2π ∫ x * f(x) dx.

What three conditions must hold for a function f(x) to be continuous at a point x = c?

1. f(c) is defined. 2. lim [x -> c] f(x) exists. 3. lim [x -> c] f(x) = f(c).

State the Rank-Nullity Theorem for an m x n matrix A representing a linear transformation T: R^n -> R^m.

rank(A) + nullity(A) = n (number of columns of A), where rank is dim(Col A) and nullity is dim(Nul A).

State the Mean Value Theorem (MVT) for a function f continuous on [a, b] and differentiable on (a, b).

There exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

State the differentiation product rule and quotient rule for differentiable functions u(x) and v(x).

Product rule: (u*v)' = u'*v + u*v'; Quotient rule: (u/v)' = (u'*v - u*v') / (v^2).

What is the general solution to ay'' + by' + cy = 0 when the characteristic roots are complex conjugates r = α ± i*β?

y(x) = e^(α*x) * [c1 * cos(β*x) + c2 * sin(β*x)].

How does the Second Derivative Test classify a critical point x = c where f'(c) = 0?

If f''(c) > 0, f has a local minimum (concave up); if f''(c) < 0, f has a local maximum (concave down); if f''(c) = 0, the test is inconclusive.

State the precise epsilon-delta definition of the limit lim [x -> c] f(x) = L.

For every ε > 0, there exists a δ > 0 such that for all x, if 0 < |x - c| < δ, then |f(x) - L| < ε.

What condition guarantees that an n x n square matrix A is diagonalizable (A = P * D * P^-1)?

A has n linearly independent eigenvectors, which always holds if A has n distinct eigenvalues or is symmetric.

How do you solve a first-order separable differential equation dy/dx = g(x) * h(y)?

Separate variables to (1 / h(y)) dy = g(x) dx, integrate both sides: ∫ (1 / h(y)) dy = ∫ g(x) dx + C, and solve for y.

What is the formula for the inverse of a 2x2 matrix A = [[a, b], [c, d]] when det(A) != 0?

A^-1 = (1 / (ad - bc)) * [[d, -b], [-c, a]].

What mathematical condition defines an exact first-order differential equation M(x, y) dx + N(x, y) dy = 0?

∂M/∂y = ∂N/∂x on a simply connected domain (equality of mixed partial derivatives).

State the differentiation chain rule for the composite function h(x) = f(g(x)).

h'(x) = f'(g(x)) * g'(x), or dy/dx = (dy/du) * (du/dx).

Calcul Différentiel et Intégral — Avancé (C1/C2) 1: Differential & Integral Calculus, Linear Algebra & Differential Equations

[Français] Calcul Différentiel et Intégral: Derivatives, integrals, Fundamental Theorem of Calculus, matrices, eigenvalues, eigenvectors, and first-order differential equations

Level: advanced
Languages: fr → fr
Cards: 32
Markdown: Calcul Différentiel et Intégral — Avancé (C1/C2) 1: Differential & Integral Calculus, Linear Algebra & Differential Equations (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	State Part 2 of the Fundamental Theorem of Calculus (evaluating a definite integral).	If f is continuous on [a, b] and F is any antiderivative of f (F' = f), then ∫[a to b] f(x) dx = F(b) - F(a).
2	State Part 1 of the Fundamental Theorem of Calculus (differentiating an accumulator function).	If f is continuous on [a, b], then g(x) = ∫[a to x] f(t) dt is continuous on [a, b], differentiable on (a, b), and g'(x) = (d/dx) ∫[a to x] f(t) dt = f(x).
3	What is the Integration by Parts formula, and what differentiation rule is it derived from?	∫ u dv = uv - ∫ v du; derived directly by integrating the product rule d(uv) = udv + v*du.
4	Under what conditions can L'Hôpital's Rule be applied to evaluate lim [x -> c] f(x) / g(x)?	When the quotient produces the indeterminate form 0/0 or ±∞/±∞ and f, g are differentiable near c; then lim f(x)/g(x) = lim f'(x)/g'(x) if the latter limit exists.
5	What is the general solution to ay'' + by' + cy = 0 when the characteristic equation has two distinct real roots r1 != r2?	y(x) = c1 * e^(r1x) + c2 e^(r2*x).
6	Define an eigenvalue λ and corresponding eigenvector v of an n x n square matrix A.	A non-zero vector v such that Av = λv; found by solving the characteristic equation det(A - λ*I) = 0.
7	What does the Wronskian determinant W(y1, y2)(x) determine about two solutions to a linear second-order ODE?	W(y1, y2) = y1y2' - y1'y2; if W != 0 at any point, the solutions y1 and y2 are linearly independent and form a fundamental solution set.
8	What is the dot product of two vectors u and v in R^n, and what geometric property does u · v = 0 indicate?	u · v = Σ [u_i * v_i] = \|\|u\|\| * \|\|v\|\| * cos(θ); u · v = 0 indicates that the non-zero vectors are orthogonal (perpendicular, θ = 90°).
9	Define a basis of a vector space V and state what the dimension of V represents.	A basis is a linearly independent set of vectors that spans V; the dimension dim(V) is the unique number of vectors in any basis of V.
10	What does the Spectral Theorem state for real symmetric matrices (A = A^T)?	Every real symmetric matrix has all real eigenvalues and can be orthogonally diagonalized: A = Q * D * Q^T, where Q is an orthogonal matrix (Q^T = Q^-1).
11	What does the Intermediate Value Theorem (IVT) state for a continuous function on [a, b]?	If f is continuous on [a, b] and u is any value between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = u.
12	What is the integrating factor μ(x) used to solve the first-order linear ODE y' + P(x)*y = Q(x)?	μ(x) = exp(∫ P(x) dx); multiplying gives (μy)' = μQ, yielding y(x) = (1 / μ(x)) * [∫ μ(x)*Q(x) dx + C].
13	What is the general solution to ay'' + by' + cy = 0 when the characteristic equation has a repeated real root r?	y(x) = (c1 + c2x) e^(r*x).
14	What is the general solution to the exponential growth/decay differential equation dy/dt = k*y with y(0) = y0?	y(t) = y0 * e^(k*t); represents exponential growth if k > 0 and exponential decay if k < 0.
15	What is the formula for the determinant of a 2x2 matrix A = [[a, b], [c, d]], and what does \|det A\| represent geometrically?	det(A) = ad - bc; \|det(A)\| represents the scaling factor by which linear transformation A multiplies areas in R^2.
16	What condition must be satisfied to multiply matrix A of dimension m x k by matrix B of dimension p x n?	The inner dimensions must be equal: k = p; the resulting matrix C = A*B has dimension m x n.
17	State the limit definition of the derivative f'(x) at a point x.	f'(x) = lim [h -> 0] (f(x + h) - f(x)) / h.
18	What is the characteristic equation for the second-order homogeneous linear ODE ay'' + by' + cy = 0 with constant coefficients?	ar^2 + br + c = 0, obtained by testing solutions of the exponential form y = e^(r*x).
19	What does it mean for a set of vectors {v1, v2, ..., vk} to be linearly independent?	c1v1 + c2v2 + ... + ck*vk = 0 holds if and only if all scalar coefficients are zero: c1 = c2 = ... = ck = 0.
20	How is the volume of a solid of revolution computed using the disk method versus the cylindrical shell method?	Disk method (revolving around x-axis): V = π ∫ [R(x)]^2 dx; Shell method (revolving around y-axis): V = 2π ∫ x * f(x) dx.
21	What three conditions must hold for a function f(x) to be continuous at a point x = c?	1. f(c) is defined. 2. lim [x -> c] f(x) exists. 3. lim [x -> c] f(x) = f(c).
22	State the Rank-Nullity Theorem for an m x n matrix A representing a linear transformation T: R^n -> R^m.	rank(A) + nullity(A) = n (number of columns of A), where rank is dim(Col A) and nullity is dim(Nul A).
23	State the Mean Value Theorem (MVT) for a function f continuous on [a, b] and differentiable on (a, b).	There exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
24	State the differentiation product rule and quotient rule for differentiable functions u(x) and v(x).	Product rule: (uv)' = u'v + uv'; Quotient rule: (u/v)' = (u'v - u*v') / (v^2).
25	What is the general solution to ay'' + by' + cy = 0 when the characteristic roots are complex conjugates r = α ± i*β?	y(x) = e^(αx) [c1 * cos(βx) + c2 sin(β*x)].
26	How does the Second Derivative Test classify a critical point x = c where f'(c) = 0?	If f''(c) > 0, f has a local minimum (concave up); if f''(c) < 0, f has a local maximum (concave down); if f''(c) = 0, the test is inconclusive.
27	State the precise epsilon-delta definition of the limit lim [x -> c] f(x) = L.	For every ε > 0, there exists a δ > 0 such that for all x, if 0 < \|x - c\| < δ, then \|f(x) - L\| < ε.
28	What condition guarantees that an n x n square matrix A is diagonalizable (A = P * D * P^-1)?	A has n linearly independent eigenvectors, which always holds if A has n distinct eigenvalues or is symmetric.
29	How do you solve a first-order separable differential equation dy/dx = g(x) * h(y)?	Separate variables to (1 / h(y)) dy = g(x) dx, integrate both sides: ∫ (1 / h(y)) dy = ∫ g(x) dx + C, and solve for y.
30	What is the formula for the inverse of a 2x2 matrix A = [[a, b], [c, d]] when det(A) != 0?	A^-1 = (1 / (ad - bc)) * [[d, -b], [-c, a]].
31	What mathematical condition defines an exact first-order differential equation M(x, y) dx + N(x, y) dy = 0?	∂M/∂y = ∂N/∂x on a simply connected domain (equality of mixed partial derivatives).
32	State the differentiation chain rule for the composite function h(x) = f(g(x)).	h'(x) = f'(g(x)) * g'(x), or dy/dx = (dy/du) * (du/dx).