Download PROBLEMS IN CALCULUS OF ONE VARIABLE BY Problems in Calculus of One Variable – I. A. – Ebook download as PDF File .pdf) or read book online. Documents Similar To PROBLEMS IN CALCULUS OF ONE VARIABLE BY I.A. MARON. 0oAULosCnuAC_Math for IIT JEE Uploaded by.

Author: Migore Monris
Country: Republic of Macedonia
Language: English (Spanish)
Genre: Health and Food
Published (Last): 17 August 2013
Pages: 53
PDF File Size: 3.93 Mb
ePub File Size: 14.11 Mb
ISBN: 876-9-25617-542-2
Downloads: 78518
Price: Free* [*Free Regsitration Required]
Uploader: Tomuro


It follows from the inequality that at any X this trinomial is non-negative. Using the Leibniz formula, find the derivatives of the indicated orders for the following functions: The full solutions developed in the text pursue two aims: The Legendre polynomial is a polynomial defined by the following formula Rodrigues’ formula: Expand the following functions: Direct Integration and the Method of Expansion Direct integration consists in using the following table of integrals: In dividing the closed interval [0, 1] into a fixed num- ber of parts we must take into consideration, in particular, two possible cases: Do calculua have a finite deriva- tive at all points of the domain?

Ascertain the existence of the following limits: Show that the inequalities 1 0.

The Differential of a Function. From the information obtained we can sketch the graph as in Fig.

The sequence of the approximations converges very slowly. On the basis of Problem 6.



Dividing the first expression by the second, we obtain: We again integrate the last integral by parts. Test the following sequences for boundedness: Solution, a Find the derivative f x: In analysing the behaviour of a function it is advisable to de- termine the following: Nonperiodic decimal fractions are called irrational numbers.

Then an infinite set of numbers x n will be found outside this neighbourhood, and that is why one cannot assert that all the numbers x m beginning with a certain one, will enter the e-neigh- bourhood of the number a. Reduction Formulas If u and t; are functions of x and have continuous derivatives, then b b J u u’ a: The sign of the function in the intervals between the zeros.

Its aim is to train the students in active approach to mathematical exercises, as is done at a seminar.

Basic Clastes of Integrable Functions 5. The polynomial is an everywhere-differentiable function.

Problems in Calculus of One Variable

Let us introduce the substitution: Estimate the absolute error. Keep in mind the following basic formulas: Inverse Functions 29 for positive values of x, hence, it increases everywhere and has an inverse function. Prove that the following cxlculus converge and find their limits: The functions are defined parametrically: Applications of the Definite Integral x-axis. If not, which of the antiderivatives can be used?


At what x will the value of F be the greatest? And so we have shown that the derivative vanishes on an infinite set of points see Fig. Applying logarithmic differentiation find the derivatives of the following functions: Subdivide the closed interval [0, 1] into n equal parts by the points x.

Let us show that the area situated above the x-axis is equal to that lying below this axis. Euler’s Substitutions Now let us expand the obtained proper rational fraction into par- tial fractions: Find the ranges of the following functions: The generic terms for points of maximum and minimum of a fun- ction are the points of extremum. Let us find the one-sided limits of the function at the point calculuz A rectangle with altitude x is inscribed in a triangle ABC with the base b and altitude h.

If a function is defined and continuous in some interval, and if this interval is not a closed one, then it can have neither naron greatest nor the least value.

Prove that between two maxima minima of a continuous ccalculus there is a minimum maximum of this function.