The Elements of the Differential Calculus: Founded on the Method of Rates Or Fluxions ...U.S. Government Printing Office, 1874 - Differential calculus |
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Common terms and phrases
1+x² abscissa ALGEBRAIC FUNCTIONS angle applying formula axis Calculus cose coseco d(x² decreasing function deduce definition denominator differen DIFFERENTIAL CALCULUS Differentials of Functions Differentiate the identity ential equation g EXAMPLES explicit function EXPONENTIAL FUNCTION feet find dy dx Find the differential formula d formula for differentiating func given instant hence hyperbola inches per second independent variable INVERSE INVERSE FUNCTIONS INVERSE TRIGONOMETRIC FUNCTIONS James Connell loga logarithmic differential m²x² miles per hour motion negative Neperian logarithm number of units numerator point moving positive quotient ratio reciprocal seco second differential sin-¹x sine sino space described square straight line tan-¹x tano tial tion Trace the curve TRANSCENDENTAL FUNCTIONS uniform rate uniform velocity units of space units per second vari variable equals variable factors variable quantity variable rate variable velocity vers-¹x whence
Popular passages
Page iii - ... described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points; superficies by the motion of lines; solids by the motion of superficies; angles by the rotation of the sides; portions of time by continual flux: and so on in other quantities. These geneses really take place in the nature of things, and are daily seen in the motion of bodies.
Page iii - I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points...
Page 3 - Variable Velocities. 15. If the velocity of a point be not uniform, its numerical measure at any instant is the number of units of space which would be described in a unit of time, were the velocity to remain constant from and after the given instant.
Page 25 - ... 17. 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 25 - ... 14. If the side of an equilateral triangle increase uniformly at the rate of 3 ft. per second, at what rate per second is the area increasing, when the side is 10 ft.
Page iii - Velocity with which they increase and are generated ; I sought a Method of determining Quantities from the Velocities of the Motions or Increments, with which they are generated ; and calling these Velocities of the Motions or Increments Fluxions, and the generated Quantities Fluents, I fell by degrees upon the Method of Fluxions, which I have made use of here in the Quadrature of Curves, in the Years 1665 and 1666.
Page 54 - The crank of a small steam-engine is i foot in length, and revolves uniformly at the rate of two turns per second, the connecting rod being 5 ft. in length ; find the velocity per second of the piston when the crank makes an angle of 45° with the line of motion of the piston-rod; also when the angle is 135°, and when it is 90°. Solution : — Let a, b and x denote respectively the crank, the connecting-rod and the variable side of the triangle ; and let 0 denote the angle between a and x.
Page 12 - J 10. A man whose height is 6 feet walks directly away from a lamppost at the rate of 3 miles an hour. At what rate is the extremity of his shadow travelling, supposing the light to be 10 feet above the level pavement on which he is walking? Draw a figure, and denote the variable distance of the man from the lamp-post by x, and the distance of the extremity of his shadow from the post by y.
Page iii - Therefore considering that quantities, which increase in equal times, and by increasing are generated, become greater or less according to the greater or less velocity with which they increase and are generated ; I sought a method of determining quantities from the velocities of the motions or increments, with which they are generated ; and calling these velocities of the motions or increments Fluxions, and the generated quantities Fluents, I fell by degrees upon the Method of Fluxions, which I have...
Page 15 - The tangent to a curve at any point is the straight line which passes through the point, and has the direction of the curve at that point* Hence, for any point of the curve, Ф denotes the inclination to the axis of x of the tangent line at that point.