ریاضی عمومیCalculus

A machine operation is error-free only when its exact result fits a normalized t-bit mantissa (leading bit 1) times a power of two.

A real cubic is bijective exactly when its derivative never changes sign, i.e. the discriminant of P' is non-positive.

The symmetric 5-point stencil approximates f'' to order h^4 using coefficients -1, 16, -30, 16, -1 /(12h^2).

The order of a fixed-point iteration equals the index of the first non-vanishing derivative of g at the fixed point.

A geometric series with r < 1 converges to (first term) / (1 - r).

Convert a radially symmetric integral over the whole plane to polar form, reducing it to a single improper p-integral in r.

If f is increasing on [a,b], then g(x)=∫_a^x f is convex, continuous, and Lipschitz, but increasing only when f≥0.

Compute an iterated double integral from the inside out, treating the outer variable as a constant during the inner integration.

Minimize the plane-distance formula over the surface, often collapsing to an algebraic inequality.

Infinite unions of closed intervals can be open and infinite intersections of open intervals can be closed, so always compute the actual set.

For a plane curve r(t)=(x(t),y(t)) , the radius of curvature is R=1/κ with κ= x'y''-y'x'' /((x')^2+(y')^2)^(3/2) .

Convert a hard closed line integral into a curl flux through any bounding surface.

A function continuous on R with finite limits at both +inf and -inf is uniformly continuous.

Fit data exactly with the unique degree-n polynomial (Lagrange/Newton) or a smooth C^2 cubic spline, and bound the polynomial error by f^(n+1)/(n+1)! times the node product.

The arc length of y=f(x) is ∫√(1+(y')^2) dx ; for the catenary y=acosh(x/a) the identity 1+sinh^2=cosh^2 erases the square root, giving length asinh(x_1/a) .

Through any n+1 points with distinct nodes there is a unique polynomial of degree ≤ n ; Lagrange and Newton are two ways to build the same polynomial.

To solve f(x)=0 , iterate x_(n+1)=x_n-(f(x_n))/(f'(x_n)) — each step replaces f by its tangent line and takes its root; near a simple root it doubles the number of correct digits per step.

Integrate by parts to express an integral with exponent n in terms of the same integral with a smaller exponent, building a recurrence you can iterate down to a base case.

Pairing geometric and arithmetic means makes one sequence climb and the other fall; AM–GM traps them between each other, so both converge to a shared limit.

The radius of a power series is 1/limsup c_n ^(1/n) — the root test that always works, even when the ratio test fails.

When an equation mixes z with bar z , Re(z) , Im(z) or z , it is not a polynomial — write z = x + iy and solve the real system.

If f is continuous and one-to-one on an interval, it must be strictly increasing or strictly decreasing — and then it maps the interval onto another interval of the same type.

Reflecting the variable across the midpoint of the interval leaves a definite integral unchanged; adding the original to the reflected copy often collapses a hopeless integrand into something trivial.

Any sequence built from a single geometric power rⁿ is decided by where r sits relative to ±1; for aₙ = rⁿ/(rⁿ+1) the limit is 0, ½, or 1 — after first checking the term is even defined.

When the unknown function sits both outside and inside an integral with a variable upper limit, differentiate both sides: the FTC turns the integral into a value of the integrand, collapsing the equation into a solvable ODE.

To find a root of a continuous function, take the supremum of the set where it is still negative — continuity squeezes the value there to exactly zero from both sides.

When a limit has a variable-limit integral ∫_0^x g(t) dt on top of a 0/0 (or ∞/∞ ) quotient, differentiate it with the FTC — its derivative is just g(x) — and the integral disappears.

Replace each piece of a 0/0 quotient by its Maclaurin series, keep terms up to the first surviving power, and read the limit off the leading coefficients.

For continuous nonnegative f on a finite interval, the n-th-root of the integral of fⁿ converges to the maximum of f — so ∫fⁿ grows like (max f)ⁿ.

An equality of two distances in the complex plane is a 'set of equidistant points' — geometrically the perpendicular bisector of the segment joining the two centers.

On an interval the sign of f' gives whether f rises or falls and the sign of f'' gives whether f' rises or falls; a one-sided monotone-bounded f' has a finite limit that equals the one-sided derivative.

A line plus a point not on it fixes a plane: build two in-plane directions, cross them for the normal, and write the point-normal equation.

A sequence given by x_(n+1)=f(x_n) converges once you show it is bounded and monotone; then the limit must be a fixed point f(L)=L.

Relative errors of a product don't simply add: δ(ab)=δ(a)+δ(b)+δ(a)δ(b) . The cross term is the catch, and it vanishes exactly when one factor is exact.

Slice a solid perpendicular to one axis, integrate the cross-sectional area A(y); for a sphere drilled through its center, the leftover ring's volume depends only on the ring's height, not the sphere's radius.

An alternating series converges if its terms shrink monotonically to 0; combine with absolute convergence to label it absolutely, conditionally, or divergent.

If a series' terms are eventually trapped between 0 and a constant multiple of a known convergent series, it converges too — the engine behind reasoning about square-summable sequences.

The condition number measures how much a linear system amplifies input error, costing you about log10(kappa) reliable digits.

The outward flux of a vector field through a closed surface equals the triple integral of its divergence over the enclosed solid.

At every point the gradient points toward the fastest increase, and its magnitude is exactly that maximum rate of change.

The sector swept from the origin along the unit hyperbola x=cosh t, y=sinh t has area exactly t/2 — the geometric meaning of the parameter.

When a system implicitly defines several variables in terms of others, differentiate the equations in place and solve a small linear system (Jacobian) for the partials — no explicit formulas needed.

A monotone function can only have jump discontinuities, and the set of those jumps is always at most countable — never an interval, never uncountable.

How to transform a second-order PDE under a linear change of variables: build the first-order operators from the chain rule, apply them twice to get the second derivatives, and substitute to simplify (e.g. the wave equation into canonical form).

To find [x^n] of an integral with a variable upper limit, expand the integrand as a power series, then bump each exponent up by one and divide: the x^n coefficient is c_(n-1)/n.

For a solid with rotational symmetry, switch to cylindrical coordinates ( dV=r dz dr dθ ) and integrate ρ r^2 to get the moment of inertia I_z about the axis.

Split [a,b] into n equal panels of width h=(b-a)/n and sum trapezoids: the interior nodes count twice, the two endpoints once, giving (h/2)[f0 + 2(f1+...+f_(n-1)) + fn].

If f is differentiable on [a,b], then f' takes every value between f'(a) and f'(b) — even though f' need not be continuous. So a derivative can never have a jump discontinuity.

The ratio (sin((2n+1)x)/2)/(sinx/2) collapses into the clean sum 1+2∑_(k=1)^ncos kx — making any integral of it trivial.

When the limits a, b of a definite integral are free, the integral is largest when [a, b] is exactly the set where the integrand is nonnegative — so set the limits at consecutive roots of f.

To glue two formulas at a breakpoint so the result is n-times continuously differentiable, force the two pieces to agree in value and in every derivative up to order n at that point — that single requirement chains all the lower orders along with it.

Use f' to split the line into monotone pieces, then count exactly one root per piece whose endpoints straddle zero — no factoring required.

Revolving a curve sweeps a surface whose area is ∫ 2π(radius) ds , where ds is arc length — pick the variable so the radius is the distance to the axis.

The mass of a solid is M=iiint_V ρ dV — integrate the (possibly variable) density over the region; when ρ is constant this collapses to ρcdotvolume .

Build the integral from sup/inf over partitions: the lower integral pushes up from below, the upper pushes down from above. They always satisfy lower ≤ upper, and f is integrable exactly when they are equal.

A function into a discrete codomain like Z is continuous exactly where it is locally constant, so its continuity set is open and its discontinuity set is closed.

Sum series like sum n/(2n+1)! by writing the numerator as a combination of the factorial's argument, so each term collapses onto a known e, sinh, or cosh series.

Split a rational function into simple poles 1/(x-r) — each differentiates n times to a tidy (-1)^n n!/(x-r)^(n+1), giving the n-th derivative with no product-rule mess.

For bounded sequences, limsup of a sum is at most the sum of limsups, and liminf is at least the sum of liminfs.

Integrate a ratio of two linear sine-cosine expressions by splitting the numerator into a multiple of the denominator, a multiple of the denominator's derivative, and a constant — turning the integral into a constant, a log, and a standard half-angle integral.

A rapidly oscillating factor like sin(nx) or cos(nx) averages any fixed integrable function to zero, so ∫_a^b f(x)sin(nx) dx → 0 as n→∞ . When f' is continuous, integration by parts proves it and gives the Θ(1/n) decay rate.

Two continuous functions that agree on a dense set (like Q ) are equal everywhere; one continuous function with a constant value on a dense set is that constant.

Integrals of the form 1/(x·√(x²±a²)) are antidifferentiated by the reciprocal substitution x = 1/t, which collapses them to ±1/√(1±t²) — an inverse-hyperbolic (or arcsin) form. The improper version converges because the tail decays like 1/x².

A limit of f(x,y) as (x,y)→(a,b) exists only if it equals the same value along every path to (a,b) . Finding two paths that give different values proves the limit does not exist.

Area swept by a polar curve is the sum of thin circular sectors, giving A = ½∫ r² dθ; for a region between two curves integrate ½(r_outer² − r_inner²), and for an overlap take the smaller radius on each ray.

A quadrature rule ∫_a^b f ≈ Q has degree of precision d if it is exact for every polynomial of degree ≤ d but not for degree d+1 . Taylor-expanding both sides about a point and matching terms reveals d and the leading error: the first mismatched power gives an error ∝ h^(d+2) f^((d+1))(ξ) .

A sum whose number of terms grows with n is squeezed by replacing every term with the largest and smallest term; if both bounds (each n copies of a single term) share a limit, the sum has that limit too.

The area of the piece of a surface z=f(x,y) lying above a region R in the xy-plane is the double integral over R of √(1+f_x²+f_y²) — the local stretch factor between the surface and its shadow.

The set of points at distance zero from A is exactly A , so it is always closed, contains A , and equals A iff A is closed.

For nonnegative integrands, domination decides convergence: if 0≤ f≤ g then ∫ g<∞ forces ∫ f<∞ , and ∫ f=∞ forces ∫ g=∞ .