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Is the function even, odd, periodic? Introduction to Mathematical Analysis 1. The process is continued until the required accuracy is obtained, which is judged of by the length of the last obtained segment. Using these values of w, find the corresponding values of x calcluus solving the equations: 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 the shape of its graph rise, fall, convexity, concavity, etc.

Determine the relative error of the result which depends on the inaccuracy in reading the angle cp.

Is the function bounded and what are its minimum and ma- ximum values? This means, by definition, that the number a is not the limit of the given sequence. Find the average value l m of alternating current intensity over time interval from to — see Problem 7. Using the L’Hospital rule find the calcuulus of the following functions: J jc 3 arc tan jc dx.

Problems in Calculus of One Variable

The best results will be obtained when the ja, having mastered the theoretical part, immedia- tely attacks the unsolved problems without referring to the text solutions unless in difficulty.


The graph of the function is shown in Fig. The Definite Integral the tabular form: Therefore the L’Hospital rule is applicable: It should be borne in mind that the function E x is defined as the maximum integer n contained in the num- iw x, i.

Arrange them in the ascending order: Investigating Functions and Sketching Graphs 3. Let the functions f x and y x satisfy all the conditions of the Cauchy theorem on the interval [a, b].

Full text of “Problems In Calculus Of One Variable by I. A. Maron”

In practice, the substitution is usually performed with the aid of monotonic, continuously differentiate functions. Thus, the total error due to using the trapezoidal formula and rounding off the ordinates does not exceed 0. Introduction to Mathematical Analysis The properties of absolute values are: Show that the equation has one root on the interval [1, 2]. In particular, the antiderivative must be a function continuous on the whole interval [a, b], A discontinuous function used as an antiderivative will lead to the wrong result.

Find y” for the following functions: Integration of Other Transcendental Functions 5. Basic Classes of Iniegrable Functions No. Compute the integral 2 dx VT- Solution.

This circumstance allows us to calculate the integrals of the in- dicated type using the method of indefinite coefficients, the essence of which is explained by fhe following example. On equivalence of what infinitesimals is it based? Prove that any polynomial with positive coefficients, which is an even function, is concave everywhere and has only one point of minimum. Find y xx if: Calculation of Limits of Functions 61 1.

Investigate the following functions for differentiability: Find the maxima and minima of the following functions: Now apply the Rolle theorem once again to the function f’ x on the intervals [ — 1, gj, [ x1]. Show that the following functions have no finite derivati- ves at the indicated points: Write the first several terms of the sequence a n. What are the domains of definition of the following functions: Direct Integration and the Method of Expansion Solution.



Is it possible to use both an- tiderivatives for computing the definite integral by the Newton- Leibniz formula? Let the point x ] be the limit point of the domain of definition X of the function f x. Find out at which of the points x the rate of change of the function is minimal. Again we have an integral of the form I. This allows us to single out the superior power of x and divide both the nume- rator and denominator by this power of x.

Can one assert that the square of a discontinuous func- tion is also a discontinuous function? Indeed, the first assertion follows x from the definition of the function arc cos x 9 while the second one can be proved in the following way.

Listed below are infinitesimal functions: Transform the integrand in the following way: