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Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. [46]: 508 The indefinite integral, also known as the antiderivative, is the inverse operation to the derivative.
This means that the upper and lower sums of the function f are evaluated on a partition a = x 0 ≤ x 1 ≤ . . . ≤ x n = b whose values x i are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [ x i , x i +1 ] where an interval with a higher index lies to the right of one ...
The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. [1] The process of finding a derivative is called differentiation.
The global maximum of x √ x occurs at x = e. Steiner's problem asks to find the global maximum for the function =. This maximum occurs precisely at x = e. (One can check that the derivative of ln f(x) is zero only for this value of x.) Similarly, x = 1/e is where the global minimum occurs for the function
if it is zero, then x could be a local minimum, a local maximum, or neither. (For example, f(x) = x 3 has a critical point at x = 0, but it has neither a maximum nor a minimum there, whereas f(x) = ± x 4 has a critical point at x = 0 and a minimum and a maximum, respectively, there.) This is called the second derivative test.
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
In fact computability can itself be defined via the lambda calculus: a function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F(x)=y if and only if f x = β y, where x and y are the Church numerals corresponding to x and y, respectively and = β ...
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