differentiable
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1: 18.32 OP’s with Respect to Freud Weights
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►where is real, even, nonnegative, and continuously differentiable.
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2: 1.4 Calculus of One Variable
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►When this limit exists is differentiable at .
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►When , is continuously
differentiable on .
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Mean Value Theorem
… ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …3: 1.6 Vectors and Vector-Valued Functions
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►when is continuously differentiable.
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►The curve is piecewise differentiable if is piecewise differentiable.
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►For , , and continuously differentiable, the vectors
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►For and twice-continuously differentiable functions
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4: 4.12 Generalized Logarithms and Exponentials
5: 2.3 Integrals of a Real Variable
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►converges for all sufficiently large , and is infinitely differentiable in a neighborhood of the origin.
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►If, in addition, is infinitely differentiable on and
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►assume and are finite, and is infinitely differentiable on .
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►Assume that again has the expansion (2.3.7) and this expansion is infinitely differentiable, is infinitely differentiable on , and each of the integrals , , converges at , uniformly for all sufficiently large .
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(a)
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On , and are infinitely differentiable and .
6: 1.13 Differential Equations
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►Let satisfy (1.13.14), be any thrice-differentiable function of , and
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1.13.18
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1.13.19
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1.13.21
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1.13.22
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7: 1.5 Calculus of Two or More Variables
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►The function is continuously differentiable if , , and are continuous, and
twice-continuously differentiable if also , , , and are continuous.
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1.5.6
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►If is continuously differentiable, , and at , then in a neighborhood of , that is, an open disk centered at , the equation defines a continuously differentiable function such that , , and .
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►If is times continuously differentiable, then
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►Sufficient conditions for validity are: (a) and are continuous on a rectangle , ; (b) when both and are continuously differentiable and lie in .
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8: 2.8 Differential Equations with a Parameter
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►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to .
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►Again, and is on .
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►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to .
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►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to .
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►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to .
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9: 18.38 Mathematical Applications
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►In consequence, expansions of functions that are infinitely differentiable on in series of Chebyshev polynomials usually converge extremely rapidly.
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