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The Definite Integral The error R in this formula is estimated as follows: II L Differential Calculus: 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: Estimate the absolute error in this root.

The given functions are functions of functions, or su- perpositions of functions, i.

Prove that between two maxima minima of a continuous function there is a minimum maximum of this function. Just a moment while we sign you in to your Goodreads account.

Problems in Calculus of One Variable

We have cos x cos 2a: Applying the L’Hos- pital rule, compare these quantities. Let us introduce the substitution: Now find the sign of the second derivative at each critical point: A system of equations for the coefficients can also be obtained by substitu- ting suitably chosen numerical values of x into both sides of the identity. The change in the limits of integration is conveniently expressed in Ch.


Solution, a Find the derivative maorn x: We assume that during each subinterval of time the n to body moves uniformly with a markn equal to its velocity at the beginning of this interval, i.

By a direct check we ascertain that they are the roots of the given equation.

The integrand is a proper rational fraction. Prove that if at the point of a minimum there exists a right-side calcuus, then it is non-negative, and if there exists a left-side derivative, then it is non-positive. Taking advantage naron the rule for differentiation of a com- posite function find the derivatives of the following functions: Solution, a The functionv 1 86 Ch.

Taking all this into account, we can sketch the graph of the antiderivative see Fig.

Solution, a Apply the results of Problem 1. Calculue if the integrand is the product of a trigonometric or an exponential function and an algebraic one, then u usually denotes the algebraic function.

Since this point claculus an interior one, the third condition of the theorem is not satisfied. This is the only extremum minimum in the interval 0, oo. The derivative f x does not vanish inside the given interval [1, e].


It is called Taylor’s formula of the function f x.


The Derivative of a Function Represented Parametrically. Indicated below are the cases when the aim can be achieved with the aid of simpler substitutions: The Differential of a Function. Let us now investigate the sign of the derivative in the neighbourhood of each of these points. Investigation mwron Fund’s 3.

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

It follows from the inequality that at any X this trinomial is non-negative. Therefore there are no critical points inside the indicated interval.

Write the function V xwhere V is the volume of the solid of revolution thus obtained and x is the length of the lateral side of the triangle.