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Calculus Equations

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Quotient Rule

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If uu and vv are functions of xx, then the derivative of their quotient u(x)v(x)\frac{u(x)}{v(x)} is given by uvuvv2\frac{u'v - uv'}{v^2}. It's used to differentiate ratios of functions.

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Chain Rule

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The derivative of a composite function f(g(x))f(g(x)) is f(g(x))g(x)f'(g(x))\cdot g'(x). It's used when a function is nested inside another function.

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Product Rule

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If uu and vv are functions of xx, then the derivative of their product u(x)v(x)u(x)v(x) is given by uv+uvu'v + uv'. It's used to differentiate the product of two functions.

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Fundamental Theorem of Calculus

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Provides a connection between differentiation and integration: if ff is continuous on [a,b][a,b] and FF is an antiderivative of ff, then abf(x)dx=F(b)F(a)\int_a^b f(x)dx = F(b) - F(a). It's used to evaluate definite integrals.

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Mean Value Theorem

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If ff is continuous on [a,b][a,b] and differentiable on (a,b)(a,b), there exists some cc in (a,b)(a,b) such that f(c)=f(b)f(a)baf'(c) = \frac{f(b)-f(a)}{b-a}. It's used to show that a function takes on its mean slope at least once.

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Limit Definition of Derivative

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The derivative of ff at x=ax=a is given by limh0f(a+h)f(a)h\lim_{h\to 0} \frac{f(a+h) - f(a)}{h}. It's used as the foundational definition of a derivative, representing the instantaneous rate of change of the function.

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L'Hôpital's Rule

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If the limit limxcf(x)g(x)\lim_{x\to c} \frac{f(x)}{g(x)} results in indeterminate forms like 0/00/0 or /\infty/\infty, then under certain conditions, it equals limxcf(x)g(x)\lim_{x\to c} \frac{f'(x)}{g'(x)}. It's used to evaluate limits involving indeterminate forms.

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Integration by Parts

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The integral of the product of two functions uu and dvdv is udv=uvvdu\int u dv = uv - \int v du. It's used to integrate products of functions that are not easily integrable in their current form.

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Taylor Series Expansion

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The Taylor series for a function ff around aa is n=0f(n)(a)n!(xa)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n. It's used to approximate functions using polynomials.

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Second Derivative Test

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For a function ff, if f(c)=0f'(c)=0 and f(c)>0f''(c)>0, cc is a local minimum. If f(c)<0f''(c)<0, cc is a local maximum. It's used to determine concavity and whether a critical point is a max or min.

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Power Rule for Derivatives

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The derivative of xnx^n where nn is a constant is nxn1nx^{n-1}. It's one of the most basic rules used to differentiate powers of xx.

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Power Rule for Integration

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The integral of xnx^n where n1n \neq -1 is xn+1n+1+C\frac{x^{n+1}}{n+1} + C where CC is the constant of integration. It's used to find antiderivatives of power functions.

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Fundamental Theorem of Line Integrals

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For a gradient field F=fF=\nabla f, the line integral over a curve CC from point aa to point bb is CFdr=f(b)f(a)\int_C F\cdot dr = f(b) - f(a). It's used to evaluate line integrals in conservative fields without parameterizing the curve.

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Implicit Differentiation

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Used when a function yy is defined implicitly as a function of xx, the derivative dydx\frac{dy}{dx} is found by differentiating both sides of the equation with respect to xx and solving for dydx\frac{dy}{dx}. It's used when functions can't be easily solved for a variable.

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Divergence Theorem

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For a vector field F\vec{F} and a solid region VV with boundary surface SS, the theorem states V(F)dV=SFdS\int_V (\nabla \cdot \vec{F}) dV = \oint_S \vec{F} \cdot d\vec{S}. It's used to relate the flow of a field through a surface to the behavior of the field inside the volume.

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Stokes' Theorem

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Relates a surface integral over a surface SS in a vector field F\vec{F} to a line integral around the boundary curve CC of SS: CFdr=S(×F)dS\oint_C \vec{F} \cdot d\vec{r} = \int_S (\nabla \times \vec{F}) \cdot d\vec{S}. It's used to convert between line integrals and surface integrals.

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Curl of a Vector Field

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The curl of a vector field F=P,Q,R\vec{F} = \langle P, Q, R \rangle is ×F=(RyQz)i+(PzRx)j+(QxPy)k\nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\vec{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\vec{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\vec{k}. It measures the rotation of the field at a point.

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Gauss's Law

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For an electric field E\vec{E} and a closed surface SS, the law states SEdA=Qencε0\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}. It relates the electric flux through a closed surface to the charge enclosed by the surface.

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Green's Theorem

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If CC is a positively oriented, piecewise-smooth, simple closed curve in a plane, and DD is the region bounded by CC, then C(Pdx+Qdy)=D(QxPy)dA\oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA. It's used to relate a line integral around a simple closed curve to a double integral over the plane region it encloses.

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Arithmetic-Geometric Mean Inequality

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For non-negative real numbers a1,a2,...,ana_1, a_2, ..., a_n, the inequality states a1a2...anna1+a2+...+ann\sqrt[n]{a_1 a_2 ... a_n} \leq \frac{a_1 + a_2 + ... + a_n}{n} with equality if and only if all aia_i are equal. It's used to relate the mean of values in their product form to their sum form.

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Partial Fraction Decomposition

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A method for expressing a rational function as the sum of simpler fractions. It's used to simplify the integration of rational functions.

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Bolzano's Theorem

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If a continuous function ff has values f(a)f(a) and f(b)f(b) of opposite signs on an interval [a,b][a,b], then there exists at least one c(a,b)c \in (a,b) such that f(c)=0f(c)=0. It's used to guarantee the existence of roots in an interval.

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Intermediate Value Theorem

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If ff is continuous on [a,b][a,b] and kk is a number between f(a)f(a) and f(b)f(b), then there exists at least one c[a,b]c \in [a,b] such that f(c)=kf(c)=k. It's used to show that a continuous function takes on every value between f(a)f(a) and f(b)f(b) within the interval [a,b][a,b].

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Lagrange Multipliers

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A method to find the local maxima and minima of a function subject to equality constraints. If f(x,y,...)f(x,y,...) is the function and g(x,y,...)=0g(x,y,...)=0 is the constraint, then the extrema occur at points where f=λg\nabla f = \lambda \nabla g where λ\lambda is the Lagrange multiplier.

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