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Calculus Equations
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Quotient Rule
If and are functions of , then the derivative of their quotient is given by . It's used to differentiate ratios of functions.
Chain Rule
The derivative of a composite function is . It's used when a function is nested inside another function.
Product Rule
If and are functions of , then the derivative of their product is given by . It's used to differentiate the product of two functions.
Fundamental Theorem of Calculus
Provides a connection between differentiation and integration: if is continuous on and is an antiderivative of , then . It's used to evaluate definite integrals.
Mean Value Theorem
If is continuous on and differentiable on , there exists some in such that . It's used to show that a function takes on its mean slope at least once.
Limit Definition of Derivative
The derivative of at is given by . It's used as the foundational definition of a derivative, representing the instantaneous rate of change of the function.
L'Hôpital's Rule
If the limit results in indeterminate forms like or , then under certain conditions, it equals . It's used to evaluate limits involving indeterminate forms.
Integration by Parts
The integral of the product of two functions and is . It's used to integrate products of functions that are not easily integrable in their current form.
Taylor Series Expansion
The Taylor series for a function around is . It's used to approximate functions using polynomials.
Second Derivative Test
For a function , if and , is a local minimum. If , is a local maximum. It's used to determine concavity and whether a critical point is a max or min.
Power Rule for Derivatives
The derivative of where is a constant is . It's one of the most basic rules used to differentiate powers of .
Power Rule for Integration
The integral of where is where is the constant of integration. It's used to find antiderivatives of power functions.
Fundamental Theorem of Line Integrals
For a gradient field , the line integral over a curve from point to point is . It's used to evaluate line integrals in conservative fields without parameterizing the curve.
Implicit Differentiation
Used when a function is defined implicitly as a function of , the derivative is found by differentiating both sides of the equation with respect to and solving for . It's used when functions can't be easily solved for a variable.
Divergence Theorem
For a vector field and a solid region with boundary surface , the theorem states . It's used to relate the flow of a field through a surface to the behavior of the field inside the volume.
Stokes' Theorem
Relates a surface integral over a surface in a vector field to a line integral around the boundary curve of : . It's used to convert between line integrals and surface integrals.
Curl of a Vector Field
The curl of a vector field is . It measures the rotation of the field at a point.
Gauss's Law
For an electric field and a closed surface , the law states . It relates the electric flux through a closed surface to the charge enclosed by the surface.
Green's Theorem
If is a positively oriented, piecewise-smooth, simple closed curve in a plane, and is the region bounded by , then . It's used to relate a line integral around a simple closed curve to a double integral over the plane region it encloses.
Arithmetic-Geometric Mean Inequality
For non-negative real numbers , the inequality states with equality if and only if all are equal. It's used to relate the mean of values in their product form to their sum form.
Partial Fraction Decomposition
A method for expressing a rational function as the sum of simpler fractions. It's used to simplify the integration of rational functions.
Bolzano's Theorem
If a continuous function has values and of opposite signs on an interval , then there exists at least one such that . It's used to guarantee the existence of roots in an interval.
Intermediate Value Theorem
If is continuous on and is a number between and , then there exists at least one such that . It's used to show that a continuous function takes on every value between and within the interval .
Lagrange Multipliers
A method to find the local maxima and minima of a function subject to equality constraints. If is the function and is the constraint, then the extrema occur at points where where is the Lagrange multiplier.
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