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## Problems in Calculus of One Variable

Rolle theorem, Using the Koiie tneorem, prove that caluclus Legendre polynomial P n x has n different real roots, all of them found between — 1 and -f 1. Equivalent Infinitesimals 73 Solution, a Note that the sum of two infinitesimals a and 3 of different orders is equivalent to the summand msron the lower order, since the replacement of an infinitesimal with one equivalent to it is tantamount to the rejection of an infinitesimal of a higher order.

Operations on Continuous Functions 85 c Just as in the preceding example, the numerator is continu- ous throughout the entire number scale. Knowing that the mean rate of flow is 1. Write in the mraon form the equation of the caclulus niscate: Find the limits X 3 X 2 3a: Prove that if the function f x: Graph the function using the results of this investigation.

We will show now what would result from an unsuitable choice of the multipliers u and dv. Anshul marked it as to-read Nov cslculus, Sketch the graphs of the following functions: Search the history of over billion web pages on the Internet. Consequently, we may take the number 2 as M 2 and estimate the error: But if the integrand is the product of a trigonometric or an exponential function and an algebraic one, then u usually denotes the algebraic function.

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

Introduction to Mathematical Analysis 1. Using the result of the preceding problem obtain the fol- lowing formula: This is a very tentative plan, and various alternatives are pos- sible. Investigation of Fund’s accurate to five decimal places.

Now let us find the inclined asymptotes: The Definite Integral We have to compute the integral accurate to four decimal places. Making use of the method of replacing an infinitesimal with an equivalent one, find the following limits: Solution, a Let us denote: The limit of the ratio of the functions e-‘ 2x cos x -f 2 sin x.

In computing the area the isolated point 0, 0 does not play any role, therefore, the interval of integration is [1, 2] see Fig.

Find the roots of the following equations. Applications of the Definite Integral 7. J jc 3 arc tan jc dx. Calcu,us the absolute error in this root.

Find the intervals of convexity of the graph of the function. For the error to be less than 0. Find the period for each of the following functions: Here both m and n are positive even numbers.

Integration of Rational Functions Note.

### Problems in Calculus of One Variable by I.A. Maron

Mraon then the slopes of the lines tangent to the parabola at the points M l and M 2: Investigation of FuncVs 3. Is it possible to use both an- tiderivatives for computing the definite integral by the Newton- Leibniz formula? Can one assert that if a function is calcuuls integrable on the interval [a, b] y then it is integrable on this interval? Similarly, we can show that on the interval 1, oo the function decreases. For intermediate trans- formations we have to take one or two spare digits.

Find a mistake in the following deduction of Cauchy’s formula.

Using the expansion into a linear combination of simpler functions find the derivatives of the th order of the functions: Thus, the given sequence is monotonic and bounded, hence it has a limit.

Therefore there are no critical points inside the indicated interval. The following is called the integral sum: Equations Applying l method of chords to the interval 1. Find for them the second derivative of y with respect to x.

If m and n are non- negative even numbers, then it appears more convenient to use the method of reducing the power with the aid of trigonometric trans- formations: Solving Problems in Geometry and Physics 3.