This text is designed as a course of mathematics for higher technical schools. It contains many worked examples that illustrate the theoretical material and serve as models for solving problems. Some of the questions that are usually discussed in these chapters have been put in the third and subsequent chapters without loss of continuity. This has made it possible to take up very early the basic concept of differential calculus—the derivative— which is required in the study of technical subjects. Experience has shown this arrangement of the material to be the best and most convenient for the student.
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A das gupta's problems plus in iit mathematics. MIR - Krechmar v. Jump to Page. Search inside document. An Implicit Function and Its Differentiation. The trrational umber V2 is expressed by rational aumbers Ti and 15 to one decimal place, Tariand i: to two decimal places, Kai4"and to three decimal places, ete SEC, 2.
The absolute value or modulus of a real number x written x is a nonnegative real number that satisfies the con- ditions x ifs; ax ife 0 such that all subsequent values of the variable, after a certain one, satisfy the condition —Maxr 0. Basic Elementary Functions. The function is defined in the infi- nite interval —oo 0 anda 1. This function is defined for all values of x. Its graph is shown in Figs. Ils graph is shown in Fig. All the enumerated F trigonometric functions are periodic.
Let us give a general definition of a Periodic function. The least such number is called the period of the function; it Fig. The period of cosx is likewise 2x. Graphs of trigonometric functions are shown in Figs. Elementary Functions 2 Fig. This function is called a function of a function or a composite function.
Example 1. Fig 2. Consequently, the variable x does not approach a In the definition of a limit it is stated that if the variable approaches the limit a, then a isa constant. Definition 2. Example 3. We shall. Note 1. We may define the limit of the function f x as x—a as follows. In other words, of two points on a number scale, the subsequent one is that which is closer to the point a; at equal distances, the subsequent one is that which is to the right of the point a.
We shall here and henceforward consider that of the two values of a function, the subsequent one is that which corresponds to the subsequent value of the argument.
It is easy to prove that both definitions of the limit of a function are equivalent, Note 2. If x takes Fig. Indeed, let an arbitrary. Inequality 3 in arbitrary e, the following inequality Segue! A Function that Approaches Infinity. We shall prove that tim eo. The boundedness of a function approaching a limit is decided by the following theorem. Theorem 1. The function see Example 8, See. Let us establish a relationship that will be important later on.
Assume that 6 lima. The number a is called the base of the logarithms. Let us now establish a relationship between decimal and natural logarithms of one and the same number x. SEC, 9. Definition 1. Continuity of a-Function Example 1. The limit of the ratio of and B is nity Lis an infinitesimal Note. Example 9. Let a. Determine the points of discontinuity of the functions: Discontinuity of Second kind at Gaye rs O40, yor see 0—-0 , 0. Are hey eqavaen?
Thus, Fig. Let us write equality 3 in full. It is thus obvious that the notion of velocity of variable motion is intimately related to the concept of a: limit. It is only with the aid of the limit concept that we can determine the velocity of variable motion. AUTH2'we have. Let the argument x receive a certain increment Ax it is imma. Then the function y will receive a certain increment Ay.
If this limit exists, it is called the derivative of the given function f x and is denoted F'. The designation f' x is not the only one used for a derivative. We shall now give a no less important geometric interpretation of the derivative. To do this we must first define line tangent to a curve af a given point. We take a curve with a fixed point M, on it. Taking a point M, on the curve we draw the secant M, M, Fig.
Corresponding to these values of x and y on the curve we have the point M, x, y. Let us increase Fig. Form the ratio a. From Fig. The secant M,M, will turn about M, and the angle will change in Ax.
It is easy to find its slope: fo. On the basis of Example , Sec. Hi a function is differentiable at every point of some interval la, 6] or a, 6 , we say that it is differentiable over.
In other words, a function cannot have'a derivative at points of discontinuity. To convince ourselves of this, let us examine several cases Example 1. A function f x Is defined in an interval 0, 2] as follows see Fig. A function Vey the graph of which is shown in Fig.
Sx Note that formula 1 also holds true when n is fractional or negative. This will be proved in Sec. The derivative of sinx is cos. The latter result has a simple geometric interpretation. Theorem 2. A constant factor may be taken outside the deriva tive sign, i. The derivative of the sum of a finite number of diffe- rentiable functions is equal to the corresponding sum of the derivatives of these functions. VI Proof. For the values of the argument x youtotw for the sake.
Differential and Integral Calculus
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Differential And Integral Calculus
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