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In single-variable calculus, the difference quotient is usually the name for the expression (+) ... LB = P 0 = P 0 + 0Δ 1 P = P ...
Suppose that one wants to approximate the 44th Mersenne prime, 2 32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log 10 (2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 10 9,808,357 × 10 0.09543 ≈ 1.25 × 10 9,808,357. Similarly, factorials can be approximated by summing the logarithms of the ...
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log 2 (8) = 3 and 2 3 = 8. The graph gets arbitrarily close to the y-axis, but does not meet it. Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations.
Therefore, the true derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: ′ = (+) (). Since immediately substituting 0 for h results in 0 0 {\displaystyle {\frac {0}{0}}} indeterminate form , calculating the derivative directly can be unintuitive.
The finite difference of higher orders can be defined in recursive manner as Δ n h ≡ Δ h (Δ n − 1 h). Another equivalent definition is Δ n h ≡ [T h − I ] n. The difference operator Δ h is a linear operator, as such it satisfies Δ h [ α f + β g ](x) = α Δ h [ f ](x) + β Δ h [g](x). It also satisfies a special Leibniz rule:
For arbitrary stencil points and any derivative of order < up to one less than the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations [6] ( s 1 0 ⋯ s N 0 ⋮ ⋱ ⋮ s 1 N − 1 ⋯ s N N − 1 ) ( a 1 ⋮ a N ) = d !
The precise size that can be guaranteed is not known, but the best bounds known on its size involve binary logarithms. In particular, all graphs have a clique or independent set of size at least 1 / 2 log 2 n (1 − o(1)) and almost all graphs do not have a clique or independent set of size larger than 2 log 2 n (1 + o(1)). [32]
Since the difference quotient is a definition of the derivative that uses limits, I think the standard analysis view is more appropriate. Russphelan ( talk ) 00:26, 21 May 2015 (UTC) [ reply ] Or, the two approaches should be clearly separated in the article.