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PROBLEMS IN CALCULUS OF ONE VARIABLE BY I.A.MARON
Keep in mind the following basic formulas: Taking all this into con sider ation, we draw the graph see Fig. Since the required function is a polynomial, the abscis- sas of the points of inflection can only be among the roots of the second derivative.
Prove that the derivative of a differentiate even function calclus an odd function, and the derivative of an odd function is an even function. It should be borne in mind that the function E x is defined as the maximum integer n contained in jn num- ber x, i.
Are the following functions identical? This interval can be narrowed still further: Therefore, henceforward we write only the necessary transfor- mations, leaving to the reader the task of checking whether the conditions of their applicability are fulfilled. Therefore, the points of extremum can only be roots of the deriva- tive.
Problems in Calculus of One Variable by I.A. Maron
For convenience and pictorialness let us variavle the following table, where all the points of interest are arranged in an ascending order: Proceeding from the definition of the limit of a function after Cauchy i.
Find the marno of decrease and increase for the follo- wing functions: Among what points should we look for points at which the composite function may have no derivative? The book comprises of chapters on differentiation of function, application of differential calculus, integration, definite integrals and improper integrals. Computing Areas in Rectangular Coordinates Solution. I know what I am talking about. The volume V of a sphere of radius r is equal to 4 yjir 3.
Consequently, the points x h Ch. A monotonic bounded sequence has a finite limit. Applying the L’Hospital rule, find the limits of the follo- wing functions: Anuj marked it as to-read Mar 14, Hence, an ohe function exists and in- creases.
Additional Problems Chapter IV. Using the methods of differential calculus, we can now carry out a more profound and comprehensive study of various properties of a function, and explain calcuulus shape of its graph rise, fall, convexity, concavity, etc. Then each of them will satisfy the conditions of Lagrange’s theorem as well.
Problems in Calculus of One Variable
Give an example of a non-monotonic function that has an inverse. There is no need to introduce an arbitrary constant after calculating each integral as is done in the above example.
In analysing the behaviour of a function it is advisable to de- termine the following: Solution, The result is a priori wrong: It is easy to show that the points x n with odd num- bers concentrate about the point 0, and the points x n with even numbers, about the point 1. The velocity of motion is equal to the time derivative i. Replace each of the following infinitesimals with an equi- valent one: The solid line in Fig. Use this example to show that the derivative of a continuous function is not always a continuous function.
Proceeding from geometric reasoning, prove that: And if a sequence is unbounded, does it mean that it has an infinite limit? Introduction maroon Mathematical Analysis Note.
The function f x y continuous on the interval [a, 6], possess- es the following properties: Similarly, we can show that on the interval 1, oo the function decreases. Thus, the equation has no solution. Therefore, x is a discontinuity. Find the average value l m of alternating current intensity ca,culus time interval from to — see Problem 7. Reduce this integral to the sum of two integrals: Let us transform each of the given integrals into an integral with limits and 1 see the preceding problem.
On the markn AB of length a a point P is taken at a distance x from the end-point A.
Introduction to Mathematical Analysis Solution. Therefore, the function f x has its minimum at the point Being the only minimum, it coincides with the least value of the function cf. The differential of a function can be written in the following way: Then the number is also rational, since it is the quotient of two rational numbers. Find the intervals of convexity of the graph of the function.