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Search the history of over billion web pages on the Internet. Maron ” See other formats I. Infinitesimal and Maeon Functions. Continuity of a Function. Arithmetical Operations on Continuous Functions. Additional Problems 91 Chapter II. Successive Differentiation of Explicit Functions. The Differential calclus a Function. Basic Theorems on Differentiable Functions Evaluation of Indeterminate Forms. Solving Problems in Geometry and Physics Convexity and Concavity of a Curve.

Additional Problems Chapter IV. Basic Methods of Integration Reduction Formulas Chapter V. Integration of Certain Irrational Expressions.

## Problems in Calculus of One Variable

Integration of Other Transcendental Mqron. Statement of the Problem. Changing the Variable in a Definite Integral. Computing Areas in Rectangular Coordinates Computing Static Moments and Moments of Inertia. Determining Coordinates of the Centre of Gravity Improper Integrals of Unbounded Functions Additional Problems Answers and Hints In fond memory of my parents From the Author This textbook on mathematical analysis is based on many calclus experience of lecturing at a higher technical college.

Its aim is ix train the students in calulus approach to mathematical exercises, as is done at a seminar. Much attention is given to problems improving the theoretical background. Therefore standard computational exercises are supple- mented by examples and problems explaining the theory, promo- ting its deeper understanding and stimulating precise mathema- tical thinking. Some counter-examples explaining the need for cer- tain conditions in the formulation of basic theorems are also in- cluded.

The book is designed along the following lines. Each section opens with a concise theoretical introduction containing the prin- cipal definitions, theorems and formulas.

Then follows a detailed solution of one or more typical problems. Finally, problems with- out solution are given, which are similar to those solved but contain certain peculiarities. Some of them are provided with hints. Each chapter except Chap. IV and V closes with a separate section of supplementary problems and questions aimed at reviewing and mraon the material of the chapter.

## PROBLEMS IN CALCULUS OF ONE VARIABLE BY I.A.MARON

These sections should prove of interest to the inquiring student, and possibly also to lecturers in selecting material for class work or seminars. The full solutions developed in the text pursue two aims: The student will find the book most useful if he uses it acti- vely, that is to say, if he studies the relevant theoretical material carefully before going on to the worked-out solutions, and finally reinforces the newly-acquired knowledge by solving the problems given for independent work.

The best results will be obtained when the student, having mastered the theoretical part, immedia- tely attacks the unsolved problems without referring to the text solutions unless in difficulty. The Absolute Value of a Real Number Any decimal fraction, terminating or nonterminating, is called a real number. Periodic decimal fractions are called rational numbers.

Every rational number may be written in the form of a ratio,ka two integers p and q y and vice versa. Nonperiodic decimal fractions are karon irrational numbers. Open, closed, and half -open intervals are covered by a single term interval.

### Problems in Calculus of One Variable : I.A. Maron :

Any real number may be depicted as a certain point on the coordinate axis which is called a proper point. Introduction to Mathematical Analysis ,aron properties of absolute values are: Prove that the number 0. Solution, To prove this, it is necessary to ascertain that the given decimal fraction is not a periodic one.

Prove that any number, with zeros standing in all deci- mal mqron numbered 10″ and only in these places, is irrational. Prove that the sum of, or the difference between, a ra- tional number a and an irrational number P is an irrational number.

Consider the sum of a and p. Assume the contrary, i. Then the number is also rational, since it is the quotient of two rational numbers. Solve the following inequalities: Find out whether the following equations have any solutions: Hence, there are no solutions. This value satisfies the initial equation. Thus, the equation has no solution. Determine the values of x satisfying the following equa- lities: Determine the values of x satisfying the following equ- maon Find the roots of the following equations.

Domain of Definition 15 following way: We may also solve this equation re- writing it in the form a: Domain of Definition The independent variable x is defined by a set X of its values. The set of values of the function y x is called the range of the given function. In particular, the functions defined by the set of natural num- bers 1, 2, 3, are called numerical sequences.

They are written in the following way: Introduction to Mathematical Analysis Solution. The function f x is defined over the whole number scale by the following law: Domain of Definition 17 B Ccalculus Introduction to Mathematical Analysis Determine the coefficients a, b y c from the above system.

Find cp [xp a: A rectangle with altitude x is inscribed in a triangle ABC with the base b and altitude h. Express the perimeter P and area 5 of the rectangle as functions of x. Calculsu of Definition 19 1. Find the domains of definition of the following functions: To ensure this the following two conditions must be satis- fied: Hence, the domain of definition of the function will be the segment [1,6]. Thus, the seg- calcupus [1,4] is the domain of definition of the function. Hence, the function is defined in the interval — oo, 0.

Find the ranges of the following functions: Domain of Definition 21 e The following inequalities must be satisfied simultaneously: Let us investigate the domain of definition of the func- tion on the left side of the equation. By a direct check we ascertain that they are the roots of the given equation.

This problem shows that a study of domains of definition of a function facilitates the solution of equations, inequalities, etc. The function f x is defined on the interval [0, 1]. What are the domains of definition of the following functions: The given functions are functions of functions, or su- perpositions of functions, i.

The function f x calcuuls said to be monotonic on the set X if it pos- sesses one of calcupus four indicated properties. The function f x is said to ja bounded on the set X if it is bounded above and below. The least number T possessing this property if such a number exists is called the period of the, function f x.

In analysing the behaviour of a function it is advisable to de- termine the following: Investigation of Functions 23 1. The domain of definition of the function. Is the function even, odd, periodic? The zeros of the function. The sign of the function in the intervals between the naron. Is the function bounded and what are its minimum and ma- ximum values? The above items do not exhaust the analysis of a function, and later on their scope will be increased.