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The method of expansion consists in expanding the integrand into calcullus linear combination of simpler functions and using the linearity Ch. Find the polynomial P x of the least degree whose graph has three points of inflection: Integration of Trigonometric and Hyperbolic Functions 1. The following limits are frequently used: Integration of Rational Functions Note. Test the following functions for monotony: Reduction Formulas If u and t; are functions of x and have continuous derivatives, then b b J u u’ a: At what speed will the ship cover any required distance at the lowest cost?

Expand the following functions: A monotonic bounded sequence has a finite limit. We integrate by parts: Anshul marked it as to-read Nov 25, A successful combination of the indicated methods, prompted by experience, often allows us to simplify the process of finding the coefficients. Using the method of chords, find approximate values of the real roots of the following equations with an accuracy up to 0.

What acceleration will the body have 4 seconds after the start?

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For convenience and pictorialness let us compile the following table, where all the points of interest are arranged in an ascending order: The integral of a function positive everywhere turns out to be zero!

Again we have an integral of the form I. Find the intervals of increase and decrease for the follo- wing functions: The graph of the function is shown in Fig.

We will distinguish three kinds of asymptotes: This value satisfies the initial equation. Estimate the integral from above i’ sin x i o Solution. It is impossible to estimate the error accurately in this case. Calculud us use the same example to demonstrate the applica- tion of the method of particular values.

Test the following sequences for limits: What are the domains of definition of the following functions: The Definite Integral verified by marn computing the initial integral: Compute the length of the normal.

Give a geometric explanation to these facts. Find the derivative with respect to x of the following functions: Find the derivative y x of the following implicit functions: If the equation has no roots close to each other, then its roots are easily separated lne this method.

Solution, a The function is defined and continuously differen- tiate throughout the entire number scale. Let us show that the area situated above the x-axis is equal to that lying below this axis. On the interval [0, a], the given function is posi- tive; consequently, the antiderivative increases.

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

Basic Classes of Integrable Functions 5. Find the -2 mistake and explain it. Thus, the curve has two vertical asymptotes: This is so-called universal substitution. Additional Problems 97 1.

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Here we have an example callculus a periodic function which does not exist in the infinite set of intervals.

Problems in Calculus of One Variable

Solution, a The domain of definition of the ome is the entire number scale — oo, oo. Therefore, x is a discontinuity. Find the inverse of the given functions: Applications of the Derivative b By the condition of perpendicularity 3a: Computing Areas in Rectangular Coordinates Solution. The volume V of a sphere of radius r is equal to 4 yjir 3. Determine the values of x satisfying the following equa- lities: Infinitesimal and Infinite Functions.

Differentiating both members of the varlable equation with respect to t calcuus get the relation between the rates of change of the coordinates: The variety of problems is good too!! Prove that the sum of, or the difference between, a ra- tional number a and an irrational number P is an irrational number. As is known, the functions x, r 1x b.

Sketch the graph f x. Replace each of the following infinitesimals with an equi- valent one: