calculus

A detailed Summary of calculus

A function, in mathematics, is used to indicate the relationship between two or more quantities. For example, the term function can be used to designate a power xn of a variable x. In addition, it can be applied to various geometric aspects of a curve, such as its slope. The concept of a function is also related to the Dirichlet concept. Dirichlet regarded y = x2 - 3x + 5 as a function which is thought of as the rule that determines y for a given x of an ordered pair of the function; thus, the preceding rule determines (3, 5), (-4, 33) as two of the infinite number of elements of the function. A function as a variable y, called the dependent variable, having its values fixed or determined in some definite manner by the values assigned to the independent variable x, or to several independent variables x1, x2,..., xk. The values of both the dependent and independent variables were real or complex numbers. The statement y = f (x), read "y is a function of x," indicated the dependence between the variables x and y; f (x) is given as a specific formula, such as f (x) = x2 - 3x + 5, or by a rule stated in words, such as f (x) is the first integer larger than x for all x's that are real numbers. Graphically, there exists even functi

lim xac f (x) = L iff for each e > 0, there exists d > 0

k = f (x0 + h)-f (x0) k/h = f (x0 + h)-f(x0) / h

ons where f (-x)= f (x) for all x ? domain (f) and odd functions where f (-x) = -f (x) for all x ? domain (f) and is symmetric about the origin. Through the usage of the vertical line test, a curve on a plane is the graph of a function if the line does not intersect the curve in more than one point.

such that if 0 < [x-c] < d then |f (x) - L| < e.

When studying the concept of limits described above, a formal understanding of the theory of differentiation can be attained. Understanding differentiation graphically: Let the variable y be a function of the independent variable x, expressed by y = f (x). If x0 is a value of x in its domain of definition, then y0 = f (x0) is the corresponding value of y. Let h be a real number (Dx, "delta x,"), and let y0 + k = f (x0 + h);

For example, the function f (x) = 1/x approaches the number 0 as x becomes positively infinite. It is important to note that a limit, as just presented, is a two-way concept: A dependent variable approaches a limit as an independent variable approaches a number or becomes infinite. The limit concept can be extended to a variable that is dependent on several independent variables. The statement "u is an infinitesimal" meaning "u is a variable approaching 0 as a limit," found in a few present-day and in many older texts on calculus, is confusing and should be avoided. Further, it is essential to distinguish between the limit of f (x) as x approach x0 and the value of f (x) when x is x0, that is, the correspondent of x0. For example, if f (x) = sin x / x, then the limit of f (x), as x approaches 0, is equal to 1, however, no value of f (x) corresponding to x = 0 exists, because division by 0 is undefined.

Some common words found in the essay are:
Axiom Induction, , Dx2y Dx2f, AC CB, Value Theorem, Dxy Dxf, = x0, = 0, derivative respect, function continuous, rate change, = x2, x =, independent variable, lim xa0, x0 +, = x0 +, x0 derivative respect, x = x0, addition function continuous, 1 lim xa0,

Approximate Word count = 2379
Approximate Pages = 10 (250 words per page double spaced)

Category: Science