A Collection of Examples of the Applications of the Differential and Integral Calculus |
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1+x² a+bx angle assume asymptote axis bx² Calc circle co-ordinates consequently corresponding cubical parabola curve cycloid deduced determined diameter Diff differential coefficients differential equation dx dy dx² dy dx dy² eliminating ellipse equal equation becomes Euler examples formula hyperbola hypocycloid infinite integral Lagrange Let dy Let the primitive logarithms Maxima and Minima maximum odd number parabola partial differential equations particular solution perpendicular places of decimals point of inflexion primitive equation radius roots surface tangent tion triangle x d x y d x
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Page 186 - Quoniam enim semper sibi similem &_ eandem Spiram gignit, utcunque volvatur, evolvatur, radiet; hinc poterit esse vel sobolis parentibus per omnia similis Emblema; Simillima Filia Matri. Vel, [si rem aeternae Veritatis Fidei mysteriis accomodare non est prohibitum] ipsius aeternae generationis Filii, qui...
Page 228 - Find that point within a triangle, from which if lines be drawn to the angular points, the sum of their squares shall be a minimum.
Page 186 - ... post varias alterationes, et tandem ipsam quoque mortem, ejusdem numero resurrecturae symbolum; adeo quidem, ut si Archimedem imitandi hodienum consuetude obtineret, libenter spiram hanc túmulo meo juberem incidi cum Epigraphe : Eadem numero mutata resurge.
Page 232 - To find a point within a triangle from which if lines be drawn to the angular points the sum of their squares is the least possible.
Page 186 - Spiram gignit, utcunque volvatur, evolvatur, radiet; hinc poterit esse vel sobolis parentibus per omnia similis Emblema; Simillima Filia Matri. Vel, [si rem aeternae Veritatis Fidei mysteriis accomodare non est prohibitum] ipsius aeternae generationis Filii, qui Patris veluti imago, &_ ab illo ut Lumen a Lumine emanans, eidem (5Juooi5ffiof existit, qualiscunque adumbratio.
Page 444 - Liouville's edition, p. 79). The equation solved is that of surfaces formed by the motion of a straight line which is always parallel to a given plane, and always passes through two given curves. 7. In the above examples V is equal to 0, and this always facilitates the application of Monge's method.
Page 172 - Location of Storage Tanks. It is usually required that diesel fuel oil storage tanks be separated from each other by a distance equal to the diameter of the largest, and from the nearest property line by the same distance.
Page 454 - To find the maximum cylinder, that can be cut from a cone whose altitude is h, and the radius of whose base is r.
Page 166 - A, as axes of co-ordinates, the equation to the cissoid is y* =- - ' The curve may be constructed mechanically. The area of the space included between the two branches and their asymptote, is equal to three times the area of the generating circle. If, instead of a circle, we employed any other curve as the generating curve, the curve generated in the same way as the C. of D., is called dssoidal.
Page 254 - Ъе the thread of a screw whose axis is coincident with the axis of z. .The thread of a screw, or the curve called the helix, is formed by a thread wrapped round the surface of a right cylinder, so as always to make the same angle with the axis ; or if the base of a right-angled triangle coincide with the base of the cylinder, and the triangle be wrapped round the cylinder, the hypothenuse will form the helix A P. To find the equations to the helix, Let the centre of the cylindrical base be the...