An Elementary Treatise on the Differential and Integral Calculus, with Numerous Examples
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1883 edition. Excerpt: ... Find in the line joining the centres of two spheres, the point from which the greatest portion of spherical surface is visible. The function to be a maximum is the sum of the two zones whose altitudes are AD and ad; hence we must find an expression for the areas of these zones. Put CM = E and cm = r, Cc = a and CP = x. The area of the zone on the sphere which has R for its radius (from Geometry, or Art. 194) = 2ttrad = 27rR--2: rBCD = 27T (R2-1, and in the same way for the other zone, from which we readily obtain the solution. Rf Rt + r 24. Find the altitude of the cylinder inscribed in a sphere of radius r, so that its whole surface shall be a maximum. CHAPTER IX. TANGENTS, NORMALS AND ASYMPTOTES. 100. Equations of the Tangent and Normal--Let P, (x', y.) be the point of tangency; the equation of the tangent line at (x1, y') will be of the form (Anal. Geom., Art. 25) y--y'--a x--x'), (1) in which a is the tangent of the angle which the tangent line makes with the axis of x. It was shown in' Article 56a that the value of this tangent is equal to the derivative of the ordinate of the point of tangency, with respect to x, is the equation of the tangent to the curve at the point (x', y'), x and y being the current co-ordinates of the tangent. Since the normal is perpendicular to the tangent at the point of tangency, its equation is, from (2), Rem.--To apply (2) or (3) to any particular curve, we substitute for---, or-=--, its value obtained from the equation of the curve and expressed in terms of the co-ordinates of the point of tangency. EXAMPLES. 1. Find the equations of the tangent and normal to the ellipse ay + VW = a52. 101. Length of Tangent, Normal, Subtangent, Subnormal, and Perpendicular on the Tangent from the Origin. Let PT...
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- 13 Sep 2013
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