Graphical Calculus (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1905 Excerpt: ...slope is /'P2 The meaning of the double sign is thus rendered evident. Each of these two branches approach the lines SA, but it is clear they never actually touch it in finite spice; for dv at the point S, --is infinitely great, and though by taking fix the point P near to S we can make the distance of the point P' from the line SA as small as we please, yet the ordinate of P' becomes enormously great, and the actual co-ordinates of the point S' would be (1, t-.) When a line and a curve have this relation to one another, i.e. the curve continually approaches as near as we please to the line, but never actually meets it in finite space, the line is said to be an "asymptote"1 to the curve. These asymptotes are of great importance in the general tracing of curves. In general, both co-ordinates of the point of contact are infinite. B.C. of Cos-1..--It is clear that by moving the vertical curve downwards through a distance =-, so that the point S is on the line OX, we shall obtain the curve y-cos'1 x, since we obtain the curve Y = cos X by moving the horizontal curve to the left through the same distance. Now, it is clear that this does not in the least alter the derived curve, so that--) _ /(sin ' x) dx J1-x the same expression as before. 1 From three Greek words, signifying "not falling together." It will, of course, be seen that neither of the curves y = sin-1 nor y = cos-1 x can have an abscissa 1 or--1; for there is no possible angle which has a sine or cosine 1 or--r. The same thing may be seen in the equation to the derived curve; for if x becomes greater than 1, say 2, we have--an imaginary expression, for it is impossible that a negative quantity should have a real square root, since the square of any real quantity, positive or n...