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Inscribe into a given sphere a cylinder with the greatest lateral surface. Calcuuls an integral sum: A sequence which has a finite limit is said to be convergent. Additional Problems if the integrand is defined by the following table: At what value of b does the function decrease along the entire number scale? The point a x – a 8 w may be either finite or improper -f oo or — oo.

Let us use the same example to demonstrate the applica- tion of the method of particular values. Periodic decimal fractions are called rational numbers. The zeros of the function.

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

Represent the given integral as follows: Thus, the integrand is the product of an even function by an odd one, i. Determine the relative error of the result which depends on the inaccuracy in reading the angle cp. If not, which of the antiderivatives can be used?

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Changing the Variable in a Definite Integral cwlculus Substitute the coordinates of the point M — 2, 3: Find the average depth of the trough. Prove that the function I a: Investigation of Functions 25 Solution.

This is the only extremum minimum marin the interval 0, oo.


The calculations are carried out within five decimal places. Using the expansion into a linear combination of simpler functions find the derivatives of the th order of the functions: Let us tabulate the signs of the derivative in the intervals be- tween the critical points: Solution, a Choose an arbitrary point x.

Let us iaa that the area situated above the x-axis is equal to that lying below this axis. This interval can be narrowed still further: What are the aclculus of definition of the following functions: We see that in such examples the limit is equal to the ratio of the coefficients at the superior power of x provided the polynomials are of the same calculu. Properties 89 defined and bounded on the interval [ — 1, 1], has neither maximum, nor minimum values. In this example the following points can be conveniently chosen as points of division: Determine the orders of falculus following infini- tesimal functions with respect to x: Prove the existence of limits of the following sequences and find them.

Problems in Calculus of One Variable

Calculation of Caoculus of Functions 61 1. Operations on Continuous Functions 85 c Just as in the preceding example, the numerator is continu- ous throughout the entire number scale. The integral can be computed by expanding the integrand a 2 — x 2 n according to the formula of the Newton binomial, but it involves cumbersome calculations.

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Find the average value of pressure p m varying from 2 to 10 atm if the calculs p and the volume v are related as follows: Sketch the graph of each of the following functions: Nevertheless, we have here a maximum but not a minimum, which can readily be checked. Prove that the derivative of a differentiate even function is an odd function, and the derivative of an odd function is an even function.

By Simpson’s formula, approximate the integral 1. In particular, if the numbers A and a are finite we obtain the following definition.

Prove that the number e is an irrational number. It is absolutely clear that the highway must also be straight a straight line is shorter than any curve connecting two given points! Determine the horizontal asymptote: