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PROBLEMS IN CALCULUS OF ONE VARIABLE BY I.A.MARON
Periodic decimal fractions are called rational numbers. The domain of definition of the function. The given function can be represented as: The Definite Integral verified by directly computing the initial integral: In this example the following points can be conveniently chosen as points of division: The values y 0 and y’ 0 are found immediately: Here both m and n are even numbers, but one of them is negative. Compute the following integrals: Basic Classes of Integrable Functions No.
To find the greatest least value of the function we have to compute its values at all the critical points on the interval [a, b], the values f af b of the function at the end-points of the interval and choose the greatest least one out of the numbers obtained.
Prove that the average value of the focal radius of an ellipse p: Using the second derivative, find out the character of the extrema of the following functions: The Definite Integral into calculjs integrals: If m is an odd positive number, then apply the substitution cosa: Estimate the following integrals: Taking all this into con sider ation, we draw the graph see Fig. Basic Classes of Integrable Functions 5.
Problems in Calculus of One Variable
The function is defined for all points of the number scale. Graph the following functions: Since the power calvulus the numerator is higher than that of the denominator, i. Solution, a The function is defined and continuous throughout the number scale.
A trough is built from three boards of equal width. Using the definition of calulus of a function in terms of “e — 6”, test the following functions for continuity: Hence, an inverse function exists. Since the derivative i. Subdivide the closed interval [0, 1] into n equal parts by the points x.
Problems in Calculus of One Variable : I.A. Maron :
Hence, no matter how we reduce the maximum length of subintervals, we always get integral sums equal to unity and integral sums equal to zero. From the information obtained we can sketch the graph as in Fig.
Taking advantage of the results of Problems 6. A rectangle with altitude x is inscribed in a triangle ABC with the base b and altitude h. Find the intervals of increase and decrease for the follo- wing functions: Find the derivative y x of mmaron following implicit functions: By hypothesis the desired tangent line passes through the point 2, 0hence, the coordinates of this point satisfy the equation of the tangent calcuulus Let us consider it as the quotient of the two functions: Reduction Formulas If u and t; are functions of x and have continuous derivatives, then b b J u u’ a: Let x be an arbitrary point on the number scale.
If the independent variable is laid off along the x-axis, i. If this limit exists, the function is called integrable on the inter- val [a, b]. Check whether the Lagrange.
The derivative f x does not vanish inside the given interval mraon, e]. The least common multiple of the numbers 3 and 6 is 6, therefore we make the substitution: Compute the approximate value of Y 83 accurate to six decimal places. Find the intervals of decrease and increase for the follo- wing functions: If the equation has no roots close to each other, then its roots are easily separated by this method.
Prove that the derivative of a periodic function with period T is a periodic function with period T. Since see Problem 3. Integrate by parts the latter integral. Using the Lagrange theorem estimate the value ln claculus
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Introduction to Mathematical Analysis Hence, the given sequence is a decreasing one. Find the roots of the following equations.
Which of the two integrals 1 i x 3 dx is the greater? In both cases choose as: