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Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h. The difference quotient is sometimes also called the Newton quotient [10] [12] [13] [14] (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat). [15]
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
For example, the third derivative with a second-order accuracy is ‴ () + (+) + (+) + (), where represents a uniform grid spacing between each finite difference ...
This expression is Newton's difference quotient (also known as a first-order divided difference). The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line.
The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. [49]: 61–63 Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function.
The answer to the question "How many cartons are needed to fit 45 eggs?" is 4 cartons, since = + rounds up to 4. Quotition is the concept of division most used in measurement. For example, measuring the length of a table using a measuring tape involves comparing the table to the markings on the tape.
The expression under the limit is sometimes called the symmetric difference quotient. [3] [4] A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point. If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not ...
In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. [citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.