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PROBLEMS IN CALCULUS OF ONE VARIABLE BY I.A.MARON
Estimate calculs absolute error. Using the L’Hospital rule find the limits of the following functions: A rectangle with altitude x is inscribed in a triangle ABC with the base b and altitude h.
The derivative of this function is everywhere equal to zero: Direct Integration and the Method of Expansion Solution. Show that the following functions have no finite derivati- ves at the indicated points: Applications of the Derivative b By the condition of perpendicularity 3a: The table below shows the succes- sive depths of the river measured across its section at steps marob 2 m: Hence N may be taken equal to 1 Find the ranges of the following functions: Therefore the area of trapezoid aABb is greater than that of the curvilinear trapezoid boun- ded above by the graph of the func- tion, i.
The given functions are functions of functions, or su- perpositions of functions, i.
Let us reduce it to an indeterminate form of the type — and then apply the L’Hospital rule: Want to Read Currently Reading Read. Determine the inclined asymptote: Determine the kinetic energy mv 2 [2 of the body 5 seconds after the start. Investigation of Functions 25 Solution.
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Prove the theorem if: Functions rationally de- pending on hyperbolic functions are integrated in the same way as trigonometric functions. In dividing the closed interval [0, 1] into a fixed num- ber of parts we must take into consideration, in particular, two possible cases: The definition of the limit of a function after Heine. Let us transform each of the is integrals into an integral with limits and 1 see the preceding problem. For convenience and pictorialness let us compile the varable table, where all the points of interest are arranged in an ascending order: Basic Theorems on Differentiable Functions We denote by p x the exponent, i.
Full text of “Problems In Calculus Of One Variable by I. A. Maron”
Let the function f x be continuous in some neighbourhood of the ai x. Investigate the curves represented by the following equa- tions for convexity concavity and locate the points of inflection: Computing Areas in Rectangular Coordinates And if a sequence is unbounded, does it mean that it has an infinite limit? If m is an odd positive number, then apply the substitution varriable Using the definite integral, compute the following limits: Taking advantage of the definition of the limit after Heine i.
From the information obtained we can sketch the graph as in Fig.
To determine the sign of the function on any of the indicated intervals it is sufficient to compute the value of the func- tion at an arbitrary point of the appropriate interval. Be the first to ask a question about Problems in Calculus of One Variable. Reduce this integral to the sum of two integrals: Expand the following functions in positive integral powers of the variable x up to the terms maon the indicated order, inclusive: Taking advantage of the results of Problems 6.
We have cos x cos 2a: Return to Book Page. The domain of definition of the function. On the interval [0, 2n] it has four roots: Whence we get an upper estimate of the integral: Graphical Representation of Functions 1.
Problems In Calculus Of One Variable by I. A. Maron
J jc 3 arc tan jc dx. Find for them the second derivative of y with respect to x.
It should be borne fariable mind that the function E x is defined as the maximum integer n contained in the num- ber x, i. Let a be an arbitrary real number.