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The lengths of the four indicated segments are expressed by the following formulas: And so, the proved equality is an equality of areas of two symmetrical curvilinear trapezoids.

Prove the existence of limits of the following sequences and find them. In how many seconds and at what distance from the ground will the body reach the highest point?

Determine the orders of the following infini- tesimal functions with respect to x: Substituting and S- 2 in the equation of the curve, we find the corresponding ordinates: Integration of Rational Functions Note. Additional Problems Chapter IV. Find the -2 mistake and explain it. Using the definite integral, compute the following limits: Make sure that a formal change of the variable t β€” x h 2 leads to the wrong result in the integral J Y x 2 dx.

Rewrite the following functions in the form of a sum of an even and an odd function: On the basis of Problem 6. Since ,v is an arbitrary point on the number scale, we have proved continuity of the function for all values of x. Test the function for continuity, find out the discontinuities and their character. Therefore the graph is a polygonal line see Fig. Passing to the limit from the integral sums, compute the integral by subdividing the interval [1, 4]: Find the intervals of increase and decrease for the follo- wing functions: The calculations are carried out within five decimal places.

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Nevertheless, we have here a maximum but not a minimum, which can readily be checked. We will distinguish three kinds of asymptotes: Can one replace infinitesimal summands by equivalent infinitesimals in computing a limit?

Represent the given equation in the form x β€” 0.

Points of Inflection 3. Each of the two vertical poles OA and CD is equipped with an electric lamp of luminous marn i fixed at a height h.

The limit of the function y x is found directly: Note that the function under consideration can be expressed through the Dirichlet function see Problem 1. It is easy to show that the points x n with odd num- bers concentrate about the point 0, and the points x n with even numbers, about the point 1.

Here we have obtained the prin- cipal properties of the logarithm proceeding only from its determi- nation with the aid of the integral. Solution, a Recall that the angle between two curves at the point of their intersection is defined as the angle formed by the lines tan- gent to these curves and drawn at this point.

Integration of Other Transcendental Functions Solution. Solve the following inequalities: What dimensions the base radius and height should the reservoir have so as to ensure the least possible expenditure of the material? claculus

Method of Tangents Newton’s method. In the first case the integral sum is equal to unity, in the second to zero.

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PROBLEMS IN CALCULUS OF ONE VARIABLE BY I.A.MARON

Using the L’Hospital rule find the limits of the following functions: In particular, if the numbers A and a are finite we obtain the following definition. Find a mistake in the following reasoning. Knowing that the mean rate of flow is 1. Determine the intervals of increase and decrease for the following functions: Let the sequences x n and y n diverge.

If in the expression 1 we substitute sin x 2 cos 2 x β€” 1 β€”sin x for sinx, then the fraction will change its sign. 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.

Let us return to x: First, let us make a draw- r is 44 ing see Fig. The velocity of rectilinear motion of a body is proporti- onal to the square root of the distance covered sas, for example, in free fall of a body.

Introduction to Ja Analysis transformation may be recommended: We will prove only the right inequality the left one is proved analogously.

Open, closed, and half -open intervals are covered by a single term interval. Differentiating this power function with respect to the intermediate argument sin xwe obtain sin 3 a: