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Indeed, the first assertion follows x from the definition of the function arc cos x 9 while the second one can be proved in the following way. Evaluation of Indeterminate Forms.

Problems in Calculus of One Variable

In practice, the substitution is usually performed with the aid of monotonic, continuously differentiate functions. Reduction Formulas 4. Solution, a We integrate by parts. Solution, a The function is defined and continuous throughout the number scale, therefore the curve has no vertical asymptote.

Assume the contrary, mron. It means that the function reaches 12 3 its minimum value but never p. First determine the value of t corresponding to the gi- ven values of x and y.

Approximating Definite Integrals 1.

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

Find the greatest and the least values of the following functions on the indicated intervals: Computing Static Moments and Moments of Inertia. Find the intervals of convexity of the graph of the function.


Prove the existence of limits of the following sequences and find them. Let us find out the character of disconti- nuities at these points.


Basic Classes ja Integrable F uncttons 5. If a sequence has an infinite limit, does it mean that this sequence is unbounded? For two infinitesimal functions to be equivalent it is necessary and sufficient that their difference be an infinitesimal of a higher order as compared with each of the two.

Hence, its discriminant is non-positive, i. The domain of definition of the function. Prove that the function I a: Applying the L’Hos- pital rule, compare these quantities. Give an example of a func- tion discontinuous everywhere whose square is a continuous function.

Make the substitution Whence a: A tank with a vertical wall of height h is installed on a horizontal plane. Furthermore, the point P cannot lie either to the left of the point A or to the right of the point B. Let the functions a x and P x be infinitesimal as x a. Since the integrand is odd, we conclude at once that the integral equals zero. As a result of n measurements of an unknown quantity x the numbers x lt x 2.

Find the absolute and relative errors allowed when replacing the increment of the function with its maroj. This method is called the method of comparison of coefficients. Furthermore, the derivative of a polynomial is a polynomial.


Using the Leibniz formula give the derivatives of the in- dicated orders for the following functions: Maron ” See other formats I.

In this mwron it is advisable to integrate with respect to y and take advantage of the symmetry of the figure see Fig. The solution of this problem is reduced iaa finding the intervals in which the derivative preserves its sign. The polynomial of the least degree with roots — 1, 0, 1 has the form ax x 2 — 1. Introduction to Mathematical Analysis Hence, the given sequence is a decreasing one.


Prove that the Dirichlet function X x see Problem 1. The derivative f x does not vanish inside the given interval [1, e]. Introduction to Mathematical Analysis Determine the coefficients a, b y c from the above system. On substituting we have: On the interval [0, 2n] it has four roots: Points of Inflection 3. Find the derivatives of the following functions: Determine the order of smallness of the quantity p with respect to the infinitesimal a.

Hence, the given function can be rewritten in the following way:

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