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Introduction to Mathematical Analysis The properties of absolute values are: The point x 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. Let us show that y x is an increasing function.

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Prove that the average value of the focal radius of an ellipse p: Sufficient Conditions for the Existence of an Extremum. Now let us find the inclined asymptotes: Every rational number may be written in the form of a ratio,of two integers p and q y and vice versa. The set of values of the function y x is called the range of the given function. Indicate the points at which the following functions have no finite derivative: Listed below are infinitesimal functions: Check whether the Lagrange.

Find the derivative sin x x The critical points are: The principal linear part of this increment A x Ax is called the diffe- rential and is denoted df x or dy. Let us show that the area situated above the x-axis is equal to that lying below this axis.

Find y” for the following functions: 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.

Therefore the function is bounded but never reaches its upper and lower bounds. Convexity and Concavity of a Curve.

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

The definition of the limit after Heine is conveniently applied when we have to prove that a function f x has no limit. Here both m calculys n are positive even numbers.

In practice, the substitution is usually performed with the aid of monotonic, continuously differentiate functions. Do the following functions have the greatest and the least values on the indicated intervals? Reduction Formulas If u and t; are functions of x and have continuous derivatives, then b b J u u’ a: Let us consider it as the quotient of the two functions: Determine then the slopes of the lines tangent to the parabola at the points M l and M 2: Therefore, solving the equations of the parabolas for x, we have: Proceeding from the definition of the limit of a function after Cauchy i.

Improper Integrals of Unbounded Functions Given the integral — x 2 dx. In dividing the closed interval [0, 1] into a fixed num- ber of parts we must take into consideration, in particular, two possible cases: For the integral JX J sin xdx find the upper and lower integral sums corresponding to the division of the closed interval [0, n] into 3 and 6 equal subintervals.

Prove the following inequalities: By hypothesis the desired tangent line passes through the point 2, 0hence, the coordinates of this point satisfy the equation of the tangent line: Since y x is a continuous function in the interval — oo, oothe inverse function x is a single-valued and continuous function of y. Therefore, we put tan a: Let us denote the radius of the circle by x y and the length of the arc by y see Fig.

Testing Sequences for Convergence 51 1. Thus, the function under consideration can be written in the following way: Solution, a The function is defined and continuous throughout the number scale, therefore the curve has no vertical asymptote. These roots are interior points of the considered interval of va- riation of the parameter t. 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.

Solution, a Recall that the angle between two curves at the point calculs their intersection is defined as the angle formed by the lines tan- gent to these curves and drawn at this point. Using the definite integral, compute the following limits: Changing the Variable in a Definite Integral. We have H 7. For each of the following functions find the points of discontinuity and determine their character: Hence, its discriminant is non-positive, i.