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In the open intervals — 00, 11, 33, 00 the function is continuous. Check whether the Lagrange.

## Problems in Calculus of One Variable

Basic Classes of Integrable Functions I. Graph the following calcklus Hence, an inverse function exists. Prove that the derivative of a periodic function with period T is a periodic function with period T. Hence, there are no solutions. Let us introduce the substitution: Note that the function under consideration can be expressed through the Dirichlet function see Problem 1. The least number T possessing this property if such a number exists is called the period of the, function f x.

Again we have an integral of the form I. Here we have obtained the prin- cipal properties of the logarithm proceeding only from its determi- nation with the aid of the integral. Find the roots of the following equations.

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

The equation of the normal is of the form: This is the only extremum minimum in the interval 0, oo. Investigate and graph the following functions: The volumetric expansion coefficient of a body is con- sidered to be approximately equal to the triple coefficient of linear expansion. The segments AT, AN are called the subtangent and the subnormal, respectively; and the o lengths MT and MN are the so-called segment of the tangent and the segment of the normal, respectively see Fig.

Determine the order of smallness of the chord of an infinitely small circular arc relative to the sagitta of the same arc. Since differentials of higher orders do not possess the property of invariance, to find d 2 y we have to consider the following two cases.

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

The values y 0 and y’ 0 are found immediately: The indicated integral can also be evalua- ted by the trigonometric substitutions: Calculation of Limits of Functions 61 1. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts or the formula for multiple integration by parts: The upper estimate of the absolute value of the polynomial P x divided by x b can be obtained as the sum of moduli of its separate terms.

Hence, the function is odd. Investigate the function defined by the definite integral X 6.

Domain of Definition 17 B Find: Reduction Formulas If u and t; are functions of x and have continuous derivatives, then is b J u u’ a: Find the domains of definition of the following functions: At what slope should the lateral boards be placed to ensure the largest cross-sectional area of the trough?

The zeros of the function. Ascertain the existence of the following limits: Basic Clastes of Integrable Functions 5.

Thus, two conditions of the Rolle theorem are satisfied. Basic Theorems on Differ entiable Functions or.