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Some counter-examples explaining the need for cer- tain conditions in the formulation of basic theorems are also in- cluded. The function is odd, its graph is symmetrical about the origin, therefore it is sufficient to investigate the function on the interval [0, oo. Show that the average value of the areas of the rectangles constructed on maeon segments AP and a 2 PB is equal to -g. Approximating Definite Integrals Estimate the error in the result obtained.

But symmetrical figures have equal areas which are expressed by definite integrals. Solution, a We integrate by parts.

It is called Taylor’s formula of the function f x. Find a mistake in the following evaluation of the integral: With the aid of the principle of substitution of equiva- lent quantities find the limits: Since the integrand is odd, we conclude at once that the integral equals zero.

Cal- culate this root approximately to within two decimal places. Prove that the number e is an irrational number. Prove that all roots of the Chebyshev-Laguerre polynomial are positive. Find the one-sided limits of the following functions as X — 0: Then a and b are found as the abscissas of the points of intersection of the indicated curves Fig. Write the first several terms of the sequence calculuz n.

Taking mafon this into con sider ation, we draw the graph see Fig. In some cases such li- mits can be found with the aid of the definite integral if it is pos- sible to transform the given sum into an integral sum. Anshuman rated it really liked it Jun 26, Is the function even, odd, periodic? Basic Theorems on Different iable Functions Solution. Additional Problems 97 1.

Applying the generalized formula for integration by parts, find the following integrals: Make the substitution Whence a: Prove that any number, with zeros standing in all deci- mal places numbered 10″ and only in these places, is irrational.

Find the inclined asymptotes: What must the radius of the circle be if we wish to have a flower-bed of the greatest possible surface area? In all these formulas the variable u is either an independent variable or a differentiable function of some variable. Ehsan Eqlimi marked it as to-read Jan 23, Thus, the side of the cube was reduced by 0.

## Problems in Calculus of One Variable

Give an example of a func- tion discontinuous everywhere whose square is a continuous function. The point a x – a 8 w may be either finite or improper -f oo or — oo. Find the one-sided limits of the functions: Integrate by parts the latter integral. Every rational number may be written in the form of a ratio,of two integers p and q y and vice versa. Taking advantage of the test for the constancy of a func- tion, deduce the following formulas known from elementary math- ematics: In this example the following points can be conveniently chosen as points of division: This value satisfies the initial equation.

Find the following integrals: Integrals of the forms I to III can be reduced to integrals of expressions rational with respect to sine or cosine ordinary or hy- perbolic by means of the following substitutions: Prove that the following sequences converge and find their limits: Determine the average value of the electromotive force E m over one period, i.

Prove that the function I a: Divyam Jain marked it as to-read May 04, Then an infinite set of dalculus 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. Transform the sum in parentheses in the following way: It means that the function reaches 12 3 its minimum value but never p. The least number T possessing this property if such a number exists is called the period of the, function f x.

Find the approximate value of: Here, of course, we assume that all derivatives and integrals appea- ring in this formula exist. Now we will show that czlculus limit of the ratio of derivatives exists: Among what points should we look for points at which the composite function may have no derivative?