Find the inverse of the function y if: In what regions will these inverse functions be defined? Find the domains of definition of the given implicit functions. Sec, 2. Calculations are best done by a slide rule. Construct the graph of the function Solution. Construct the graphs of rational integral functions of degree two parabolas. Find the points ol intersection of this pa- rabola with the Ac-axis. Construct the graphs of the exponential and logarithmic func- tions: Construct the graphs of the functions: Construct the graph of the function obtained.
Express the area of the rectangle y as a func- tion of the base x. Plot the graph of this function and find its greatest value. Lfmits 1. The limit of a sequence. Example 1. The limit of a function. One-sided limits. Example 2. For which values of n will we have the inequal- ity e is an arbitrary positive number? Find the limits: A similar procedure is also possible in many cases for fractions contain- ing irrational terms. Example 3. I Example 4. Find lim Solution. Example 5. Example 6. Example 7.
Find Solution. Example 9. Hmfl -. Prove that Solution. Regard it as the limit of the corresponding finite fraction. Find the limit of the perimeters of regular n-gons inscribed in a circle of radius R and circumscribed about it as n - oo. M n inscribed in a logarithmic spiral Introduction to Analysis [Ch. Show that the limit of the perimeter of the bro- ken line thus formed dilTers from the length of AB despite the fact that in the limit the broken line "geometrically merges with the segment AB".
The side a of a right triangle is divided into n equal parts, on each of which is constructed an inscribed rectangle Fig. Infinitely Small and Large Quantities 1. Infinitely small quantities infinitesimals. The sum of two infinitesimals of different orders is equivalent to the term whose order is lower. The limit of a ratio of two infinitesimals remains unchanged if the terms of the ratio are replaced by equivalent quantities. By virtue of this theorem, when taking the limit of a fraction lim!
Infinitely large quantities infinites. As in the case of infinitesimals, we introduce the concept of infinites of different orders. For what values of x is the ine- quality! Calculate numeri- cally for: a e Determine the order of smallness of: a the surface of a sphere, b the volume of a sphere if the radius of the sphere r is an infinitesimal of order one.
What will the orders be of the radius of the sphere and the volume of the sphere with respect to its surface? Let the central angle a of a cir- cular sector ABO Fig. Prove that the length of an infinitesimal arc of a circle of constant radius is equivalent to the length of its chord. Can we say that an infinitesimally small segment and an infinitesimally small semicircle constructed on this segment as a diameter are equivalent? Using the theorem of the ratio of two infinitesimals, find Compare the values obtained with tabular data.
Continuity of Functions 1. Definition of continuity. If a function is continuous at every point of some region interval, etc. Points of discontinuity of a function. For continuity of a function f x at a point JC Q , it is necessary and suf- ficient that 38 In t reduction to Analysis [Ch. I Example 3. At all other points this function is, obviously, continuous. Discontinuities of a function that are not of the first kind are called discontinuities of the second kind.
Infinite discontinuities also belong to discontinuities of the second kind. Properties of continuous functions. A function f x continuous in an interval [a, b] has the following proper- ties: 1 f x is boanded on [a, 6J, i. Prove that the rational integral tunction is continuous for any value of x. Prove that the rational fractional function is continuous for all values of x except those that make the de- nominator zero. Prove that if the function f x is continuous and non- negative in the interval a, 6 , then the function is likewise continuous in this interval.
Prove that the function y cos x is continuous for any x. Plot the graph of this function. Prove that the absolute value of a continuous function is a continuous function. Investigate the following functions for continuity: Investigate the following functions for continuity and construct their graphs: Give an example to show that the sum of two discontin- uous functions may be a continuous function.
Show that the equation has a real root in the interval 1,2. Approximate this root. Prove that any polynomial P x of odd power has at least one real root. Prove that the equation has an infinite number of real roots. Calculating Derivatives Directly 1. Increment of the argument and increment of the function. Example t. For the function Sec. V ' 1 1 J 7 Solution. The derivative. One-sided derivatives. Infinite derivative. Define: a the mean rate of rotation; b the instantaneous rate of rotation. A hot body placed in a medium of lower temperature cools off. What is to be understood by: a the mean rate of cooling; b the rate of cooling at a given instant?
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What is to be understood by the rate of reaction of a sub- stance in a chemical reaction? Calculate f' 8 , if Find the angle be- tween these tangents. Tabular Differentiation 1. Basic rules for finding a derivative. Table of derivatives of basic functions I. Rule for differentiating a composite function. Find the derivative of the function Solution. Find the derivatives of the following functions the rule for differentiating a composite function is not used in problems Algebraic Functions s 5 QO ,.
Inverse Circular and Trigonometric Functions Exponential and Logarithmic Functions Hyperbolic and Inverse Hyperbolic Functions Composite Functions In problems to , use the rule for differentiating a composite func- tion with one intermediate argument. O CL F F Sec. Construct the graphs of the functions y and y'. Find ' if Prove that the derivative of an even function is an odd function, and the derivative of an odd function is an even func- tion.
Prove that the derivative of a periodic function is also a periodic function. Taking logarithms we get In y v In u. Find y' , if 2 Solution. The derivative of an inverse function. Find the derivative x y , if Solution. The derivatives of functions represented parametrically. The derivative of an implicit function. Find the derivative y x if 0. Prove that a function represented parametrically by the equations satisfies the equation Find y' at the point A!
Geometrical and Mechanical Applications of the Derivative 1. Equations of the tangent and the normal. The straight line passing through the point of tangency perpendicularly to the tangent is called the normal to the curve. The angle between curves. Using a familiar formula of analytic geometry, we get 3. Segments associated with the tangent and the normal in a rectangular coordinate system. The tangent and the normal determine the following four Sec 4] Geometrical and Mechanical Applications of the Deriiative 61 segments Fig. Segments associated with the tangent and the normal in a polar sys- tern of coordinates.
Find the equations of the tangent and the normal to the curve at the point 1,0. Will we have the same thing at 2,4? Show that the hyperbolas intersect at a right angle. At the point 1,2 evaluate the lengths of the segments of the subtangent, subnormal, tangent, and normal. What procedure of construction of the tan- gent to the ellipse follows from this? Find the angle between the tangent and the radius vector of the point of tangency in the case of the logarithmic spiral A Fig. The law of motion of a material point thrown up at an angle a to the horizon with initial velocity V Q in the vertical plane OXY in Fig.
Find the trajectory of motion and the distance covered. Also determine the speed of motion and its direction. What is the rate of change of its ordinate when the point passes through 5,2? At what rate are the area of the surface of the sphere and the volume of the sphere increasing when the radius becomes 50 cm? A nonhomogeneous rod AB is 12 cm long. Find the mass of the entire rod AB and the linear density at any point M.
What is the linear density of the rod at A and S? Derivatives of Higher Orders 1. Definition of higher derivatives. Leibniz rule. Higher-order derivatives of functions represented parametricaKy. OOU 8. Find the velocity and the ac celeration of motion of M,. What is the velocity and the acceleration of M l at the in tial time and when it passes through the origin? What are the maximum values of the absolute velocity and th absolute acceleration of Ai,? Find 70 Differentiation of Functions [Ch. Find y" at 0,1 if Find at the point 1,1. Differentials of First and Higher Orders 1. First-order differential.
The differential of a Fig. Principal properties of differentials. Applying the differential to approximate calculations. By how much approximately does the side of a square change if its area increases from 9 m 2 to 9. Higher-order differentials. A second-order differential is the differential of a first-order differential: We similarly define the differentials of the third and higher orders. Here the primes denote derivatives with respect to M. Find the increment Ay and the differentia! Find the increment and the differential of this function and ex- plain the geometric significance of the latter.
In the following problems find the differentials of the given functions for arbitrary values of the argument and its increment. X Find the approximate value of sin Replacing the increment of the function by the differen- tial, calculate approximately: a cos 61; d In 0. Derive the approximate formula and find approximate values for j! Find the approximate value of arc sin 0. Solut ion. Mean-Value Theorems 1. Rolle's theorem.
Lagrange's theorem. Cauchy's theorem. Find the appropriate values of g. Does the Rolle theorem hold for this function on [0. Does the Rolle theorem hold for the function on the interval [0, JT]? Show that this equation cannot have any other real root. Test whether the Lagrange theorem holds for the function on the interval [2,1] and find the appropriate intermediate value of.
Evaluate the error in the formula Evaluating the indeterminate forms and. Other indeterminate forms. In certain cases it is useful to combine the L'Hospital rule with tht finding of limits by elementary techniques. Compute lim JL1 form ". Prove that the limits of X-H Find these limits directly. The Extrema of a Function of One Argument 1. Increase and decrease of tunctions.
Tlu Junction y f x is called increasing decreasing on some interval if, fo. Test the following function for increase and decrease: Solution. On a number scale we get two intervals of monot- onicity: 00, 1 and 1, -f oo. Determine the intervals of increase and decrease of the func- tion Solution. Thus, the function being tested in- creases in the interval oo, 1 , decreases in the interval 1, 1 and again increases in the interval 1, -f oo. Extremum of a function. The minimum point or maximum point of a function is its extremal point bending point , and the minimum or maximum of a function is called the extremum of the function.
More generally: let the first of the derivatives not equal to zero at the point x of the function f x be of the order k. Greatest and least values. The least greatest value of a continuous function f x on a given interval [a, b] is attained either at the critical points of the function or at the end-points of the interval [a, b]. Determine the intervals of decrease and increase of the func- 1ions: Explain the result in geometrical terms. Prove the inequalities: Separate a given positive number a into two summands such that their product is the greatest possible.
What right triangle of given perimeter 2p has the great- est area? It is required to build a rectangular playground so that it should have a wire net on three sides and a long stone wall on the fourth.
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It is required to make an open rectangular box of greatest capacity out of a square sheet of cardboard with side a by cutting squares at each of the angles and bending up the ends of the resulting cross-like figure. An open tank with a square base must have a capacity of v litres. What size will it be if the least amount of tin is used? Which cylinder of a given volume has the least overall surface? In a given sphere inscribe a cylinder with the greatest volume.
In a given sphere inscribe a cylinder having the greatest lateral surface. In a given sphere inscribe a cone with the greatest volume. Inscribe in a given sphere a right circular cone with the greatest lateral surface. About a given cylinder circumscribe a right cone of least volume the planes and centres of their circular bases coincide. Which of the cones circumscribed about a given sphere has the least volume?
A sheet of tin of width a has to be bent into an open cylindrical channel Fig. What should the central angle cp be so that the channel will have maximum capacity? Out of a circular sheet cut a sector such that when made into a funnel it will have the greatest possible capacity.
An open vessel consists of a cylinder with a hemisphere at the bottom; the walls are of constant thickness. What will the dimensions of the vessel be if a minimum of material is used for a given capacity? A point M x , lies in the first quadrant of a coordi- nate plane.
Draw a straight line through this point so that the triangle which it forms with the positive semi-axes is of least area. Inscribe in a given ellipse a rectangle of largest area with sides parallel to the axes of the ellipse. A messenger leaving A on one side of a river has to get to B on the other side. Knowing that the velocity along the bank is k times that on the water, determine the angle at which the messenger has to cross the river so as to reach B in the shortest possible time. The width of the river is h and the distance be- tween A and B along the bank is d.
A lamp is suspended above the centre of a round table of radius r. At what distance should the lamp be above the table so that an object on the edge of the table will get the greatest illumination? The intensity of illumination is directly proportion- al to the cosine of the angle of incidence of the light rays and is inversely proportional to the square of the distance from the light source.
It is required to cut a beam of rectangular cross-section ont of a round log of diameter d. What should the width x and the height y be of this cross-section so that the beam will offer maximum I resistance a to compression and b to bending? A homogeneous rod AB, which can rotate about a point A Fig. A linear centimetre of the rod weighs q kilograms. Determine the length of the rod x so that the force P should be least, and find P mln. Sphere A of mass M moving with ve- locity v strikes fi, which, having acquired a certain velocity, strikes C of mass m. What mass should B have so that C will have the greatest possible velocity?
N identical electric cells can be formed into a battery in different ways by combining n cells in series and then combin- ing the resulting groups the number of groups is ] in par- allel. For what value of n will the battery produce the greatest current? Prove that the most probable value of x is the arithmetic mean of the measurements. The Direction of Concavity. Points of Inflection 1. The concavity of the graph of a function.
Points of inflection. Points at which f' x Q or f x does not exist are called critical points of the second kind. And it is not a point of inflection if the signs of f x are the same in the above-indicated intervals.
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We have bx a, Fig. We have: It is obvious that y" does not vanish anywhere. Equating to zero the denominator of the fraction on the right of 1 , we find that y" does not exist for x 2. The tan- gent at this point is parallel to the axis of ordinates, since the first derivative y' is infinite at x 2. Find the intervals of concavity and the points of inflection of the graphs of the following functions: Asymptotes 1.
Vertical asymptotes. Find the asymptotes of the curve lotos- Solution. We seek the inclined asymptotes. Testing a curve for asymp- totes is simplified if we take into consideration the symmetry of the curve. Find the asymptotes of the curve Sec. Find the asymptotes of the following curves: Graphing Functions by Characteristic Points In constructing the graph of a function, first find its domain of definition and then determine the behaviour of the function on the boundary of this domain. It is also useful to note any peculiarities of the function if there are any , such as symmetry, periodicity, constancy of sign, monotonicity, etc.
Then find any points of discontinuity, bending points, points of inflection, asymptotes, etc. These elements help to determine the general nature of the graph of the function and to obtain a mathematically correct outline of it. Construct the graph of the function Solution, a The function exists everywhere except at the points x 1. The function is odd, and therefore the graph is symmetric about the point 0 0, 0.
From the symmetry of the curve it follows that there is no left-hand asymptote either. To determine the signs of y' or, respectively, y" in each of the indicated intervals, it is sufficient to determine the sign of y' or y" at some one point of each of these intervals. Sec 4] Graphing Functions by Characteristic Points 97 It is convenient to tabulate the results of such an investigation Table I , calculating also the ordinates of the characteristic points of the graph of the function. We form a table,, including the characteristic points Table In addition io the characteristic points it is useful to find the points of intersection of 34 the curve with the coordinate axes.
Curvature A good exercise is to graph the functions indicated in Fxam- ples Construct the graphs of the following functions represented parainelrically. Differential of an Arc. Curvature 1. Differential of an arc. Curvature of a curve. V M Fig. The radius of curvature R is the reciprocal of the absolute value of the curvature, i. Sec 5] Differential of an Arc.
Curvature We have the following formulas for computing the curvature in rectan- gular coordinates accurate to within the sign : 1 if the curve is given by an equation explicitly, y f x , then 2 if the curve is given by an equation implicitly, F x, y 0, then F F F xx. Circle of curvature. The radius of the circle of curvature is equal to the radius of curvature, and the centre of the circle of curvature the centre of curvature lies on the normal to the curve drawn at the point M in the direction of concavity of the curve.
The evolute of a curve is the locus of the centres of curvature of the curve. The normal MC of the involute P 2 is a tangent to the evolute P,; the length of the arc CC l of the evolute is equal to the corresponding increment in the radius of curvature CC, M,C, AfC; that is why the involute P 2 is also called the evolvent of the curve P, obtained by unwinding a taut thread wound onto P, Fig.
To each evolute there corresponds an infinitude of invo- lutes, which are related to different initial lengths of thread. Vertices of a curve. The vertex of a curve is a point of the curve at which the curvature has a maximum or a minimum. To determine the vertices of a curve, we form the expression of the curvature K and find its extremal points. Find the vertex of the catenary Solution. Equating x a dx a M 6. Find the differential of the arc, and also the cosine or sine of the angle formed by the radius vector and the tangent to each of the following curves: Compute the curvature of the given curves at the indicated points: Find the radii of curvature at any point of the given lines: Compute the coordinates of the centre of curvature of the given curves at the indicated points: Extrema and the Geometric Applications of a Derivative [Ch.
Write the equations of the circles of curvature of the given curves at the indicated points: Find the evolutes of the curves: Prove that the evolute of the logarithmic spiral r is also a logarithmic spiral with the same pole. Direct Integration 1. Basic rules of integration. Table of standard integrals. Integration under the sign of the differential. Rule 4 considerably expands the table of standard integrals: by virtue of this rule the table of integrals holds true irrespective of whether the variable of integration is an independent variable or a differentiate function.
We took advantage of Rule 4 and tabular integral 1. We implied u jc 2 , and use was made of Rule 4 and tabular integral V. This type of transformation is called integration under the differential sign. Using the basic rules and formulas of integration, find the following in- tegrals: , V dv lUlM. J a a I LUo. J V x J cos'xdx. J sinh x J H J sinh x cosh x ' Find the indefinite integrals: Integration by Substitution 1.
Change of variable in an indefinite integral. Hencu, Sometimes substitutions of the form are used. Examples 2, 3, 4 Sec. Trigonometric substitutions. It is sometimes more convenient to make use of hyperbolic substitutions, which are similar to trigonometric substitutions see Example For more details about trigonometric and hyperbolic substitutions, see Sec. Put xlant. Applying trigonometric substitutions, find the following in- tegrals: , A formula for integration by parts. In certain cases, integration by parts yields an equation from which the desired integral is determined.
Applying the formula of integration by parts, find the following integrals: Jarcsin A-rfjc. Applying various methods, find the following integrals: Standard Integrals Containing a Quadratic Trinomial 1. To perform the transformations in 1 , it is best to take the perfect square out of the quadratic trinomial. The methods of calculation are similar to those analyzed above. Find dx Solution. We put whence 4. Integration of Rational Functions t. The method of undetermined coefficients. Integration of a rational function, after taking out the whole part, reduces to integration of the proper rational fraction where P x and Q A- are integral polynomials, and the degree of the nume- rator P x is lower than that of the denominator Q A-.
These coeffi- cients may likewise be determined by putting [in equation 2 or in an equi- valent equation] x equal to suitably chosen numbers second method. Here, 5 and A lt B lt.. The Ostrogradsky method. If Q A has multiple roots, then P x A' v p r. Y A "- 6 where Q, A: is the greatest common divisor of the polynomial Q x and its derivative Q' A- ; X A- and Y x are polynomials with undetermined coefficients, whose degrees arc, respectively, less by unity than those of Q, A- and Q 2 x.
The undetermined coeflicients of the polynomials X x and Y x are computed by differentiating the identity 6. Tjpfc Intagrating Certain Irrational Functions 1. Integrals of the f.
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The coefficients of the polynomial Q n -i x and the number K are found by differentiating identity 3. Integrals of the form i; dx a n V ax They are reduced to integrals of the form 2 by the substitution: A a Find the integrals: Integrals of the binomial differentials x t 5 where m, n and p are rational numbers. Chebyshev's conditions. The integral 5 can be expressed in terms of a finite combination of elementary functions only in the following three cases: 1 if p is a whole number; 2 if " is a whole number.
Here, use is made of the substitution Indefinite Integrals Ch.
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Hence, we have here Case 2 integrability. Find the integrals: We do the same if n is an odd positive number. Integrating Trigonometric Functions 1. Integrals of the form : 'm.. C x x J J cos aA: f Example 8. In individual cases, it is useful to apply artificial procedures see, for example, I39 o-. Integration of Hyperbolic Functions Integration of hyperbolic functions is completely analogous to the inte- gration of trigonometric functions.
J sinh 2 A: cosh 2 A; Sec. V Integration of Various Transcendental Functions Find the integrals: J jc arc cos 5jc 5 Using Reduction Formulas Derive the reduction formulas for the following integrals: C- 1 Varc sinyd. J cos lnx dA:. The Definite Integral as the Limit of a Sum 1. Integral sum. Geometrically, S,, is the algebraic area of a step-like figure see Fig. The definite integral. Geometrically, the definite integral 2 is the algebraic sum of the areas of the figures that make up the curvilin- ear trapezoid aABb, in which the areas of the parts located above the -axis are plus, those below the jc-axis, minus Fig.
What is the lim S n Solution. Here, Ax. Whence Hence Fig. Summing, we get the area of the step-like figure Fig. S and g are constant. A definite integral with variable upper limit. The Newton-Leibniz formula. Find the integral a 3' -1 ' 4 Solution. Applying the Newton-Leibniz formula, find the integrals: 1 X J Jdt. L 4 Jl 2 Improper Integrals 1. Integrals of unbounded functions.
Integrals with infinite limits. Test for convergence the elliptic integral dx Sec. Evaluate the improper integrals or establish their divergence : Test the convergence of the following integrals: 1 Applying the indicated substitutions, evaluate the following integrals: Evaluate the following integrals by means of appropriate substitutions: Prove that if f x is an even function, then Sec. Show that! Using the formula obtained, Devaluate I 9 and 7 Mean-Value Theorem 1. Evaluation of integrals. Evaluate the integral 5L 2 M Solution. The mean value of a function.
Determine the signs of the integrals without evaluating them: n b Definite Integrals [Ch. Prove that r dx lies between 4 0. Find the exact value of this integral. Evaluate the integrals: n 4 J gqpjj. Integrating by parts, prove that JI Sec. The Areas of Plane Figures 1.
Area in rectangular coordinates. The sought-for area is expressed by the integral b X Definite Integrals Ch. Find the area of the ellipse Fig. Due to the symmetry, it is sufficient to compute the area of a quadrant and then multiply the result by four. The area in polar coordinates. Find the area contained inside Bernoulli's lemniscate Fig. Find the entire area bounded by the astroid Compute the area contained within the curve Find the entire area of the cardioid r a 1-f coscp.
Find the area bounded by Pascal's limagon Fig. The Arc Length of a Curve 1. The arc length in rectangular coordinates. Differentiating the equation of the astroid, we get Sec. The arc length of a curve represented parametrically. Fig 49 Fig. Find the length of one arc of the cycloid Fig. The entire curve is described by a point as cp ranges from to 3ji. Sec, 9] Volumes of Solids Find the length of the evolute of the ellipse Find the length of the first turn of Archimedes' spiral Volumes of Solids '. The volume of a solid of revolution. If the curve is defined in a different form parametrically, in polar coor- dinates, etc.
Computing the volumes of solids from known cross-sections. Determine the volume of a wedge cut off a circular cylinder by a plane passing through the diameter of the base and inclined to the base at an angle a. The radius of the base is R Fig. Find the volume of a solid formed by rotation, about the x-axis, of an area bounded by the x-axis and the parabola A right parabolic segment whose base is 2a and altitude h is in rotation about the base. Determine the volume of the result- ing solid of revolution Cavalieri's "lemon". Find the volume of an obelisk whose parallel bases are rectangles with sides A, B arid a, ft, and the altitude is h.
Find tfte volume of the solid formed by these squares. Find the volume of the solid generated by the circle. The plane of a moving triangle remains perpendicular to the stationary diameter of a circle of radius a. The base of the triangle is a chord of the circle, while its vertex slides along a straight line parallel to the stationary diameter at a distance h from the plane of the circle. Find the volume of the solid called a conoid formed by the motion of this triangle from one end of the diameter to the other.
Zfta Fig. The dimensions of a parabolic mirror AOB are indicated in Fig. It is required to find the area of its surface. Find the area of the surface of a spindle obtained by rotation of a lobe of the sinusoidal curve ys'mx about the -axis. Centres of Gravity. Guldin's Theorems 1. Static moment. In a similar manner we define the static moment of a system of points relative to a plane. For the cases of geometric figures, Itie density is considered equal to unity. II] Moments. Moment of inertia. In the case of a continuous mass, we get an appropriate integral in place of a sum. Find the moment of inertia of a triangle with base b and altitude h about its base.
For the base of the triangle we take the x-axis, for its altitude, the y-axis Fig Divide the triangle into infinitely narrow horizontal strips of width dy t which play the role of elementary masses dm. Centre of gravity. The coordinates of the centre of gravity of a plane figure arc or area of mass M are computed from the formulas where MX and My are the static moments of the mass. In the case of geomet- ric figures, the mass M is numerically equal to the corresponding arc or area. Centres of Gravity Guldin's Theorems Solution.
We have and Whence f, ' ,. Guldin's theorems. Theorem 1. The area of a surface obtained by the rotation of an arc of a plane curve about some axis lying in the. Theorem 2.
The volume of a solid obtained by rotation of a plane figure about some axis lying in the plane of the figure and not intersecting it is equal to the product of the area of this figure by the circumference of the circle described by the centre of gravity of the figure. Find the static moments about the coordinate axes of a segment of the straight line x y o o lying between the axes. Find the centre of gravity of an arc of a circle of radius a subtending an angle 2a.
Find the coordinates of the centre of gravity of an area bounded by the curves Find the centre of gravity of a homogeneous right circular cone with base radius r and altitude h. Find the moment of inertia of a circle of radius a about its diarneler. Find the moments of inertia of a rectangle with sides a and b about its sides. Find the moment of inertia of a right parabolic segment with base 26 and altitude ft about its axis of symmetry. Find the moment of inertia of a homogeneous right circular cone with base radius R and altitude H about its axis.
Find the moment of inertia of a homogeneous sphere of radius a and of mass M about its diameter. The path traversed by a point. What is the mean velocity cf motion during this interval? The work of a force. What work has to be performed to stretch a spring 6 cm, if a force of 1 kgf stretches it by 1 cm? Whence the sought-for work is 0. Kinetic energy. Find the kinetic energy of a homogeneous circular cylinder of density 6 with base radius R and altitude h rotating about its axis with angular velocity CD. For the elementary mass dm we take the mass of a hollow cylinder of altitude h with inner radius r and wall thickness dr Fig.
Pressure of a liquid. To compute the force of liquid pressure we use Pascal's law, which states that the force of pressure of a liquid on an area S at a depth of immersion h Is where y is the specific weight of the liquid. Find the force of pressure experienced by a semicircle of radius r submerged vertically in water so that its diameter is flush with the water surface Fig Solution, We partition the area of the semicircle into elements strips parallel to the surface of the water.
The pressure experienced by this element is where y is the specific weight of the water equal to unity. At what distance from the initial position will the body be in t seconds from the time it is thrown? The velocity of a body thrown vertically upwards with initial velocity v air resistance allowed for is given by the formula where t is the time, g is the acceleration of gravity, and c is a constant.
Find the altitude reached by the body. A point on the x-axis performs harmonic oscillations about the coordinate origin; its velocity is given by the formula where t is the time and t; , co are constants. What is the mean value of the absolute magni- tude of the velocity of the point during one cycle?
Find the path covered by the point from the commencement of motion to full stop. A rocket rises vertically upwards. Calculate the work that has to be done to pump the water out of a vertical cylindrical barrel with base radius R and altitude H. Calculate the work that has to be done in order to pump the water out of a conical vessel with vertex downwards, the radius of the base of which is R and the altitude H. Calculate the work needed to pump oil out of a tank through an upper opening the tank has the shape of a cylinder with horizontal axis if the specific weight of the oil is y, the length of the tank H and the radius of the base R.
On a visit to China in , adopting a baby girl with a contingent of other families, my tour guide pulled me aside toward the end of our trip and popped a question that took me by surprise. Is there a distinctly and identifiably Jewish way of doing business? If so, should we package it up and share our centuries-old recipe with the larger world?
It started with hearing legends of my grandfather. With an eighth-grade education, he supported eight younger siblings by selling beans on the side of the road. He started several companies, two of which ended up being traded on the New York Stock Exchange. Countless American Jewish families have similar tales and hence no surprise that we are disproportionally represented among the ranks of the wealthy.
But, not always in the way the gentile world or even most Jews think.