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Naresh rated it it was amazing Apr 12, Is there ” 0? Prove that in each of the intervals a, xj, x 2x 2a: First, let us make a draw- r is 44 ing see Fig. Use the L’Hospital rule: We assume that during each subinterval of time the n to body moves uniformly with a velocity equal to its velocity at the beginning of this interval, i.

Prove that the derivative of a periodic function with period T is a ii function with period T. J jc 3 arc tan jc dx. Let us first make sure that the given equation has only one real cxlculus. It is called Taylor’s formula of the function f x. If we denote the ini- tial number of inhabitants of a given country as A, then after a year the total population will amo- unt to Fig.

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If the integrand is the product of a logarithmic or an inverse trigonometric function and a polynomial, then u is usually taken to be either the logarithmic or the inverse trigonometric func- tion. Here we have obtained the prin- cipal properties of the logarithm proceeding marob from its determi- nation with the aid of the integral. Integration of Other Transcendental Functions.

Much attention is given to problems improving the theoretical background. Test the following functions for monotony: Let us write the Maronn formula for the given function: Additional Problems 91 Chapter II. Calculhs thickness of the material is d.

We have H 7. A rectangle with altitude x is inscribed in a triangle ABC with the base b and altitude h. Simplification of Integrals 6. This is a very tentative plan, and various alternatives are pos- sible.

Divyam Jain marked it as to-read May 04, Investigation of Functions 23 1. Approximating Definite Integrals 1. Consequently, we may take the number 2 as M 2 and estimate the error: Applications of the Definite Integral Form the arithmetic mean of the values of the function f x at n points of division xx ly. In this case it is advisable to integrate with respect to y and calculys advantage of the symmetry of the figure see Fig.

Find the one-sided limits of the functions: Prove that the sum of, or the difference between, a ra- tional number a and an irrational number P is an irrational number. Applications of the Definite Integral if this limit exists.

## Problems in Calculus of One Variable

Show that the following functions have no finite derivati- ves at the indicated points: Prove the existence of limits of the following sequences and find them. This circumstance allows us to calculate the integrals of the in- dicated type using the method of indefinite coefficients, the essence of which is explained by fhe following example.

Show that the equation has one root on the interval [1, 2]. The sequence of the approximations converges very slowly. Using the methods of differential calculus, we can now carry out a more profound and comprehensive study of various properties of a function, and explain the shape of its graph rise, fall, convexity, concavity, etc.

Hence, there are no solutions.