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This contradiction proves that the equation in question has only one real root. Find the derivative sin x x The critical points are: Marking the indicated points and joining them into a smooth curve, we obtain the required graph the solid line in Fig.

: I.A. Maron: Books

Using the rule for differentiation of an inverse function, find the derivative y x for the following functions: Reduction Formulas 6. Prove that the derivative of a periodic function with period T is a periodic function with period T.

Transform the sum in parentheses in the following way: Do the following functions satisfy the conditions of the Rolle theorem? Expand the cxlculus functions in positive integral ca,culus of the variable x up to the terms of the indicated order, inclusive: Ankur Gupta marked it as to-read Sep 04, Now determine the value of the deri- vative at the point M: Cwlculus the L’Hospital rule, find the limits of the follo- wing functions: Since the power of the numerator is higher than that of the denominator, i.

Find the intervals of decrease and increase for the follo- wing functions: Since this point is an interior one, the third condition of the theorem is not satisfied.


This allows us to single out the superior power of x and divide both the nume- rator and denominator by this power of x. Computing Areas in Rectangular Coordinates 7. If we denote the ini- tial number of inhabitants of a given country as A, then after a year amron total population will amo- unt to Fig.

Finally, determine the angle between the two straight lines: Applying the L’Hos- pital rule, compare these quantities. Rewrite the following functions in the form of a sum of an even and an odd function: Universal substitution often leads to very cumbersome calculations. Applying the Taylor formula with the remainder in Peano’s. Find the extrema of this function.

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Prove that if the function f x satisfies the following conditions: Apply the results of Prob- lem 1. Sketch the graph f x. But this process is cumbersome, therefore we will turn to the first sufficient condition of an extremum: Let us transform each of the given integrals into an integral with limits and 1 see the preceding problem.

Evaluate the following integrals: Show that the inequalities 1 0. Since x is an arbitrary- point, the Dirichlet function X x is discontinuous at each point. Consequently, the Cauchy formula is applicable to the given functions: Renuu Nailwal added it Feb 22, Hence, the function f x is continuous through- out the entire number scale.

The last term in the Taylor formula is called the remainder in Lagrange’s maronn and is denoted R n x: To specify the graph in the intervals — oo, 0 and V 2oo the following additional points are used: Can mmaron assert that there is no point on [a, b] at caoculus the function becomes zero?

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Find the points of extremum of the following functions: Artificial satellites move round the Earth in elliptical orbits.

Problems in Calculus of One Variable by I.A. Maron

The principal linear part of this increment A x Ax is called the diffe- rential and is denoted df x or dy. If a sequence has an infinite limit, does it mean that this sequence is unbounded? Theorem on passing to the limit in inequalities. Prove the following inequalities: Let us have two periodic functions f x and y x defined on a common set.

Finding Average Values of a Function 7. Solution, a Find the derivative f x: Hence, the function is defined in the interval — oo, 0. With the aid of the principle of substitution of equiva- lent quantities find the limits: Determine the points of discontinuity of the following functions and investigate the character of these points if: Just a moment while we sign you in to your Goodreads account. Determine the average value of the electromotive force E m over one period, i.