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abscissa acceleration algebraic altitude angle arbitrary assumed asymptote axis become infinite called change sign coefficient constant coordinates corresponding values cosec curve is symmetrical cycloid decreasing denominator denote derivative described determine differential distance dx dy dy _ ellipse employed epicycloid equal evidently example expression factor Find the radius find the value formula fraction func given function hence hypocycloid increasing function independent variable indeterminate form initial line inverse inverse function ip dip maxima and minima maximum value minimum ordinate moving point negative obtain parabola passes perpendicular point moving point of inflexion polar equation pole preceding article primary value prove quantity radius of curvature ratio represented root semi-cubical parabola square straight line subtangent symbol takes the form tangent line Taylor's Theorem Theorem tion Trace the curve triangle value of f(x vanish velocity whence
Page i - Johnson — DIFFERENTIAL CALCULUS, an Elementary Treatise on the ; Founded on the Method of Rates or Fluxions. By JOHN MINOT RICE, Professor of Mathematics in the United States Navy, and WILLIAM WOOLSEY JOHNSON, Professor of Mathematics at the United States Naval Academy. Third Edition, Revised and Corrected. Demy 8vo.
Page 28 - A man standing on the edge of a wharf is hauling in a rope attached to a boat at the rate of 4 ft. per second. The man's hands "being 9 ft. above the point of attachment of the rope, how fast is the boat approaching the wharf when she is at a distance of 12 ft. from it? 5 ft. per second.
Page 36 - ... the opposite buildings. If a man is walking across to the opposite side of the street at the rate of 4 miles an hour, at what rate per hour does his shadow move upon the walls — (a) when he is 5 ft. from the curbstone ? (ft) when he is 20 ft. from the curbstone ? (a) 96 miles ; (ft) 6 miles. 26. Assuming the volume of a tree to be proportional to the cube of its diameter, and that the latter increases uniformly ; find the ratio of the rate of its volume when the diameter is 6 inches to the...
Page 71 - If the space described in / seconds be expressed by the formula 3. If a point moves in a fixed path so that show that the acceleration is negative and proportional to the cube of the velocity.
Page 196 - ... as n becomes infinite. The preceding proposition may be employed to determine the limits of certain expressions which occur in undetermined forms. Thus it is evident that the expression •v'l .2-- -n increases indefinitely with n, since the ratio n \/(n — 1) I increases indefinitely with n. In a similar manner it may be shown that each of the expressions Vn and v/logn approaches the limit unity as n becomes infinite.
Page 117 - 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 to cross the track of the second at the least possible distance from her ? W.
Page 14 - It appears, on combining the results expressed in equations (2), (3), and (4), that the differential of a polynomial is the algebraic sum of the differentials of its terms ; and that constant terms disappear from the result.
Page 116 - 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.
Page 36 - Two locomotives are moving along two straight lines of railway which intersect at an angle of 60° ; one is approaching the intersection at the rate of 25 miles an hour, and the other is receding from it at the rate of 30 miles an hour ; find the rate per hour at which they are separating from each other when each is 10 miles from the intersection. 2^ miles. 25. A street-crossing is 10 ft. from a street-lamp situated directly above the curbstone, which is 60 ft. from the vertical walls of the opposite...