The Elements of the Differential Calculus, Founded on the Method of Rates Or Fluxions ...U.S. Government Printing Office, 1875 - 66 páginas |
Términos y frases comunes
acceleration altitude assumed binomial theorem changes sign coefficients cone constant convergent series convex portions cos² cosh curve d²x d²y d2y dx2 d³y decreasing deduce denote determine diagram dx dx dx)² dx² dy dx employed equa equicrescent EXAMPLES Expand log expression f(xo factor Find the nth formula func given sphere gives a minimum hence IMPLICIT FUNCTIONS increasing function independent variable inscribed logarithmic logarithmic series maxima and minima maximum minimum value mum value negative nth derivative number of terms obtain odd function parabola parabolic segment particular value passes through zero primitive equation putting quantity radius rectangle remainder second derivative second differential equation sin x sin² sinh space described square successive differentials Taking the derivative Taylor's Theorem tion value of f(x value of x vanish velocity Whence
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Página 50 - When a ray of light passes from one medium to another, it is refracted so that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the velocities in the two media.
Página 53 - ... isosceles triangle that can be described about a given ellipse, having its base parallel to the major axis. The height is three times the minor semi-axis. 15. Inscribe the greatest parabolic segment in a given isosceles triangle. The altitude of the segment is three-fourths that of the triangle. 16. A steamer whose speed is 8 knots per hour and course due north sights another steamer directly ahead, whose speed is 10 knots, and whose course is due west. What must be the course of the first steamer...
Página 23 - ... weight is to be raised by means of a lever weighing n pounds per linear inch, which has its fulcrum at one end, and at a fixed distance a from the point of suspension of the weight w ; find the length of the lever in order that the power required to raise the weight may be a minimum.
Página 7 - If a point moves in a fixed path so that show that the acceleration is negative and proportional to the cube of the velocity.
Página 20 - ... two following equations : The required values of x will therefore be found among the roots of these equations. The case which usually presents itself, and which will therefore be considered first, is that in which the required value of x is a root of the equation f(x) = o. 99. As an illustration, let it be required to divide a number into two such parts that the square of one part multiplied by the cube of the other shall give the greatest possible product. Denote the given number by a, and the...
Página 43 - The lower corner of a leaf, whose width is a, is folded over so as just to reach the inner edge of the page : find...
Página 26 - The intercepts on the axes are га and ib. 15. A high vertical wall is to be braced by a beam which must pass over a parallel wall a feet high and b feet distant from the other , find the length of the shortest beam that can be used for this purpose. Take as the independent variable the inclination of the beam to the horizon (a* + **) r 6.
Página 24 - A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter, required the height and breadth of the window when the quantity of light admitted is a maximum. The radius of the semicircle must equal the height of the rectangle.
Página 32 - The illumination of a plane surface by a luminous point being directly as the cosine of the angle of incidence of the rays, and inversely as the square of its distance from the...
Página 26 - Substituting this value of x in equation (3), we have the maximum value required. 101. As a further illustration, let it be required to determine the greatest cylinder that can be inscribed in a given segment of a paraboloid of revolution. Let a denote the altitude, and b the radius of the base of the segment. The equation of the generating parabola is of the form Since (a, b) is a point of the curve, we have the condition £' = Of a ; eliminating 4<r, the equation of the curve is FIG ii.