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If f is a differentiable function on R ( or an open interval ) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero ; points where are called critical points or stationary points ( and the value of f at x is called a critical value ).
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If and f
If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).
The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.
If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.
If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).
If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.
If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from.
If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).
If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.
If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral
If and is
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.
If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If the existent form is to be retained new factors that reinforce it must be introduced into the situation.
If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??
If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
If and differentiable
If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.
If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.
If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.
If a risk averse consumer has a utility function over consumption x, and if is differentiable, then the consumer is not prudent unless the third derivative of utility is positive, that is,.
If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.
If r < 1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable.
If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇ f which takes the point a to the vector ∇ f ( a ).
If ƒ is an infinitely differentiable function defined on an open set D ⊂ R, then the following conditions are equivalent.
If each component of V is continuous, then V is a continuous vector field, and more generally V is a C < sup > k </ sup > vector field if each component V is k times continuously differentiable.
If γ: b → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L ( γ ) in analogy with the example above by
Suppose V is a subset of R < sup > n </ sup > ( in the case of n = 3, V represents a volume in 3D space ) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have
If the target function is not easily differentiable, the differential with respect to each variable can be approximated as
If is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not null.
If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function doesn't need to be differentiable for the Jacobian to be defined, since only the partial derivatives are required to exist.
If p is a point in R < sup > n </ sup > and F is differentiable at p, then its derivative is given by J < sub > F </ sub >( p ).
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