Calculus (mathematics) I INTRODUCTION Limits This graph, which charts the function f(x)=1x, shows that the value of the function approaches zero as x becomes larger and larger.

Calculus (mathematics) I INTRODUCTION Limits This graph, which charts the function f(x)=1x, shows that the value of the function approaches zero as x becomes larger and larger. Yet even as x approaches infinity, the value of the function will never quite fall to zero. Zero, therefore, is said to be the limit of this function. © Microsoft Corporation. All Rights Reserved.. Calculus (mathematics), branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis. Calculus is widely employed in the physical, biological, and social sciences. It is used, for example, in the physical sciences to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of decay of a radioactive material. In the biological sciences a problem such as the rate of growth of a colony of bacteria as a function of time is easily solved using calculus. In the social sciences calculus is widely used in the study of statistics and probability. Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, the most, the slowest, or the least. These maximum or minimum amounts may be described as values for which a certain rate of change (increase or decrease) is zero. By using calculus it is possible to determine how high a projectile will go by finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero. Many general principles governing the behavior of physical processes are formulated almost invariably in terms of rates of change. It is also possible, through the insights provided by the methods of calculus, to resolve such problems in logic as the famous paradoxes posed by the Greek philosopher Zeno. The fundamental concept of calculus, which distinguishes it from other branches of mathematics and is the source from which all its theory and applications are developed, is the theory of limits of functions of variables (see Function). Let f be a function of the real variable x, which is denoted f(x), defined on some set of real numbers surrounding the number x0. It is not required that the function be defined at the point x0 itself. Let L be a real number. The expression is read: "The limit of the function f(x), as x approaches x0, is equal to the number L." The notation is designed to convey the idea that f(x) can be made as "close" to L as desired simply by choosing an x sufficiently close to x0. For example, if the function f(x) is defined as f(x) = x2 + 3x + 2, and if x0 = 3, then from the definition above it is true that This is because, as x approaches 3 in value, x2 approaches 9, 3x approaches 9, and 2 does not change, so their sum approaches 9 + 9 + 2, or 20. Another type of limit important in the study of calculus can be illustrated as follows. Let the domain of a function f(x) include all of the numbers greater than some fixed number m. L is said to be the limit of the function f(x) as x becomes positively infinite, if, corresponding to a given positive number e, no matter how small, there exists a number M such that the numerical difference between f(x) and L (the absolute value f(x) - L) is less than e whenever x is greater than M. In this case the limit is written as 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, or bilateral, 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 approaches 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 however, no value of f(x) corresponding to x = 0 exists, because division by 0 is undefined in mathematics. The two branches into which elementary calculus is usually divided are differential calculus, based on the consideration of the limit of a certain ratio, and integral calculus, based on the consideration of the limit of a certain sum. II DIFFERENTIAL CALCULUS Derivatives The derivative of a function at a given point is equal to the slope of the line that is tangent to the function at that given point. In this example, the derivative of f(x) at x0 is defined as the slope of AB in the limit of h going to zero. As h becomes increasingly smaller, B moves along the curve towards A, and AB increasingly approximates T, the tangent to the curve at x0. © Microsoft Corporation. All Rights Reserved. Let the dependent 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 and k be real numbers, and let y0 + k = f(x0 + h ). (? x, read "delta x," is used quite frequently in place of h .) When ?x is used in place of h,? y is used in place of k. Then clearly and This ratio is called a difference quotient. Its intuitive meaning can be grasped from the geometrical interpretation of the graph of y = f(x). Let A and B be the points (x0, y0), (x0 + h, y0 + k), respectively, as in the Derivatives illustration. Draw the secant AB and the lines AC and CB, parallel to the x and y axes, respectively, so that h = AC, k = CB. Then the difference quotient k/h equals the tangent of angle BAC and is therefore, by definition, the slope of the secant AB. It is evident that if an insect were crawling along the curve from A to B, the abscissa x would always increase along its path but the ordinate y would first increase, slow down, then decrease. Thus, y varies with respect to x at different rates between A and B. If a second insect crawled from A to B along the secant, the ordinate y would vary at a constant rate, equal to the difference quotient k/h, with respect to the abscissa x. As the two insects start and end at the same points, the difference quotient may be regarded as the average rate of change of y = f(x) with respect to x in the interval AC. If the limit of the ratio k/h exists as h approaches 0, this limit is called the derivative of y with respect to x, evaluated at x = x0. For example, let y = x2 and x = 3, so that y = 9. Then 9 + k = (3 + h )2; k = (3 + h )2 - 9 = 6h + h 2; k/h = 6 + h; and Referring back to the Derivatives illustration, the secant AB pivots around A and approaches a limiting position, the tangent AT, as h approaches 0. The derivative of y with respect to x, at x = x0, may be interpreted as the slope of the tangent AT, and this slope is defined as the slope of the curve y = f(x) at x = x0. Further, the derivative of y with respect to x, at x = x0, may be interpreted as the instantaneous rate of change of y with respect to x at x0. If the derivative of y with respect to x is found for all values of x (in its domain) for which the derivative is defined, a new function is obtained, the derivative of y with respect to x. If y = f(x), the new function is written as y' or f'(x), Dxy or Dxf(x), (dy)/(dx) or df(x)/dx. Thus, if y = x2, y + k = (x + h )2; k = (x + h )2 - x2 = 2xh + h 2; k/h = 2x + h , whence Thus, as before, y' = f'(x) = 6 at x = 3, or f'(3) = 6; also, f'(2) = 4, f'(0) = 0, and f'(-2) = -4. Derivatives and Indefinite Integrals of Common Functions © Microsoft Corporation. All Rights Reserved. As the derivative f'(x) of a function f(x) of x is itself a function of x, its derivative with respect to x can be found; it is called the second (order) derivative of y with respect to x, and is designated by any one of the symbols y" or f"(x), Dx2y or Dx2f(x), (d 2y)/(dx2) or (d 2f(x))/(dx2). Third- and higher-order derivatives are similarly designated. Every application of differential calculus stems directly or indirectly from one or both of the two interpretations of the derivative as the slope of the tangent to the curve and as the rate of change of the dependent variable with respect to the independent variable. In a detailed study of the subject, rules and methods developed by the limit process are provided for rapid calculation of the derivatives of various functions directly by means of various known formulas. Differentiation is the name given to the process of finding a derivative. Differential calculus provides a method of finding the slope of the tangent to a curve at a certain point; related rates of change, such as the rate at which the area of a circle increases (in square feet per minute) in terms of the radius (in feet) and the rate at which the radius increases (in feet per minute); velocities (rates of change of distance with respect to time) and accelerations (rates of change of velocities with respect to time, therefore represented as second derivatives of distance with respect to time) of points moving on straight lines or other curves; and absolute and relative maxima and minima. III INTEGRAL CALCULUS Integration An estimate of the area under a curve can be found by adding the areas of a series of rectangles whose tops closely match the shape of the curve. As this graph shows, any rectangle will always have part above the curve and part below the curve. However, if more rectangles with thinner widths are drawn, there will be less difference between the parts above and below the curve. To find a precise area from adding the areas of rectangles, the rectangles would have to be infinitely thin. The calculus technique of integration provides a way to find the area under a curve. © Microsoft Corporation. All Rights Reserved. Let y = f(x) be a function defined for all x's in the interval [a,b], that is, the set of x's from x = a to x = b, including a and b, where a