Elements of the Differential Calculus

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. 1898 edition. Excerpt: ...units per second; and, as before, the actual rate of change at the instant denoted by t is lim AO=dO At = At dt' This is the number of current-units that would be gained in the next second if the rate of gain were uniform from the time t to the time t + 1. Since S. = Art. 21 dx dt dt hence measures the ratio of the rates of change of y dec and of x. It follows that the result of differentiating y=f(?) (i) may be written in either of the forms g =/'(), (2)!=/'()f. (3) The latter form is often convenient, and may also be obtained directly from (1) by differentiating both sides with regard to t. It may be read: the rate of change of y is f'(x) times the rate of change of x. Returning to the illustration of a moving point P, let its coordinates at time t be x and y; then measures the rate dt of change of the-coordinate, and may be called the velocity of Jp resolved parallel to the z-axis, or the-component of the velocity. Similarly, & is the-compo Fig 24. nent of velocity. These three rates of change are connected by the equation Ex. 1. If a point describe the straight line Sx + y = 5, and if x increase h units per second, find the rates of increase of y and of s. Ex. 2. A point describes the parabola y2 = 12 x, in such a way that when x--3, the abscissa is increasing at the rate of 2 feet per second: at what rate is y then increasing? Find also the rate of increase of s. Ex. 3. A person is walking towards the foot of a tower on a horizontal plane at the rate of 5 miles per hour; at what rate is he approaching the top, which is 60 feet high, when he is 80 feet from the bottom? Let x be the distance from foot of tower at time t, and y the distance from the top at the same time; then X + 602 = y dx dy dt dt When x is 80...