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Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found. Differentiate both sides of the equation with respect to time (or other rate of change). Often, the chain rule is employed at this step. Substitute the known rates of change and the known quantities into the equation.
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
Once a value of y is chosen, say a, then f(x,y) determines a function f a which traces a curve x 2 + ax + a 2 on the xz-plane: = + +. In this expression, a is a constant, not a variable, so f a is a function of only one real variable, that being x. Consequently, the definition of the derivative for a function of one variable applies:
Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point ...
In mathematics, a rate is the quotient of two quantities, often represented as a fraction. [1] If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change ...
One thinks of δF/δρ as the gradient of F at the point ρ, so the value δF/δρ(x) measures how much the functional F will change if the function ρ is changed at the point x. Hence the formula ∫ δ F δ ρ ( x ) ϕ ( x ) d x {\displaystyle \int {\frac {\delta F}{\delta \rho }}(x)\phi (x)\;dx} is regarded as the directional derivative at ...
The mean value theorem gives a relationship between values of the derivative and values of the original function. If f(x) is a real-valued function and a and b are numbers with a < b, then the mean value theorem says that under mild hypotheses, the slope between the two points (a, f(a)) and (b, f(b)) is equal to the slope of the tangent line to ...