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Problems in Calculus of One Variable
It vanishes at two points: Using the first derivative, find the extrema of the follo- wing functions: Introduction to Mathematical Analysis Solution. Let the sequence x n converge and the sequence y n diverge.
Given the integral — x 2 dx. Let us find the inclined asymptotes.
Maron ” See other formats I. Can one assert that if a function is absolutely integrable on the interval [a, b] y then it is integrable on this interval? Prove that the sum of, or the difference between, a ra- tional number a and an irrational number P is an irrational number.
Full text of “Problems In Calculus Of One Variable by I. A. Maron”
If the independent variable is laid off along the x-axis, i. Find the domains of i.a.mzron of the following functions: Prove that the product of two even or two odd functions is an even function, whereas the product of an even and an odd function is an odd function. Additional Problems 97 1.
Write the function V xwhere V is the volume of the solid of revolution thus obtained and x is the length of the lateral side of the triangle. The method of expansion consists in expanding the integrand into a linear combination of simpler functions and using the linearity Ch.
We make the substitution: Find y xx if: In particular, if the numbers A and a are caldulus we obtain the following definition. X variabls 1 X Investigate the function thus obtained for a minimum in the interval 0, oo: The point variablee is called a discontinuity of the first kind of the fun- ction f x if there exist the limits to the right and to the left and they are finite.
The Definite Integral 6. The Legendre polynomial is a polynomial defined by the following formula Rodrigues’ formula: What acceleration will the body have 4 seconds after the start?
Thus, the equation has no solution. It should be remem- bered that in sketching the graph of a function the principal refe- rence points are the points of the curve corresponding to the extremal values of the function, points of inflection, asymptotes. Reduction Formulas I.a.marpn V. Integrating by parts, derive reduction formulas for calcu- lating onee following integrals: We have H 7.
The greatest value of each summand is attained at a: Specifying the roots, i. Now return to the initial variable x: Solve all questions from arihant.
Therefore it is possible to subtract the number n from the upper and lower limits: Basic Clastes of Integrable Functions 5. Investigating Functions and Sketching Graphs Using the results of the investigation, sketch the graph of the function see Fig.
Knowing that the mean rate of flow is 1. Let us find the one-sided limits of the function at the point 1: From e and f it follows that the absolute error in log tan x is always more than that in log sin x for the same x and A x. The new limits of integration t 1 and t 2 are found from the equa- i.
Can a sum, difference, product or quotient gy irrational numbers be a rational number? It should be borne in mind that the function E x is defined as the maximum integer n contained in the num- ber x, i. How can I download I. Return to Book Page.
If the function increases decreasesthen the inverse function also increases decreases. Periodic decimal fractions are called rational numbers. Therefore the L’Hospital rule is applicable: The problems of the exis- tence of an inverse function and of expressing it analytically should not be confused. Applying the generalized formula for integration by parts, find the following integrals: