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**calculation**is a deliberate process that transforms one or more inputs into one or more results. The term is used in a variety of senses, from the very definite arithmetical**calculation**of using an algorithm, to the vague heuristics of calculating a strategy in a competition, or calculating the chance of a successful relationship between two people.The pocket-sized Hewlett-Packard HP-35 scientific

**calculator**was the first handheld device of its type, but it cost US$395 in 1972. This was justifiable for some engineering professionals but too expensive for most students. Around 1974 the handheld electronic scientific**calculator**made slide rules largely obsolete.The derivative of () = for any (nonvanishing) function f is: ′ = ′ (()) wherever f is non-zero. In Leibniz's notation, this is written (/) =.The reciprocal rule can be derived either from the

**quotient**rule, or from the combination of power rule and chain rule.Integrals are also used in thermodynamics, where thermodynamic integration is used to calculate the

**difference**in free energy between two given states. Computation Analytical. The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus.The

**difference**between a given convergent and π is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction 22 / 7 is greater than π , but 22 / 7 − π is less than 1 / 7 × 106 = 1 / 742 (in fact, 22 / 7 − π is just more than 1 / 791 = 1 / 7 × 113 ).Structural effects can also be important. The

**difference**between fumaric acid and maleic acid is a classic example. Fumaric acid is (E)-1,4-but-2-enedioic acid, a trans isomer, whereas maleic acid is the corresponding cis isomer, i.e. (Z)-1,4-but-2-enedioic acid (see cis-trans isomerism).Applicable to: m-by-n matrix A of rank r Decomposition: = where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A, which one can apply to obtain all solutions of the linear system =.

**Newton's method**is a powerful technique—in general the convergence is quadratic: as the method converges on the root, the**difference**between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method.