Elements of the Differential Calculus: With Examples and Applications |
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Common terms and phrases
2gR² A₁ angle approaches zero axis Binomial Theorem body called centre chord circle constant coördinates corresponding increments corresponding points curvature cycloid D₂ D₂u D₂y D₂y)² decreases definite value derivative descent developed diameter difference differential earth ellipse equation evolute EXAMPLES expression external area finite formulas function fx Fx fx-fa given curve given point hence higher order hn+1 hyperbola hypocycloid increases indefinitely Indeterminate Forms infinitely near points integral length limit approached limiting position locus loga logx Maxima and Minima maximum mean velocity minimum values multiplied negative normal obtained ordinate parabola perpendicular plane principal infinitesimal radii radius radius of curvature ratio rectangle secant line second order secx sin² sinx Suppose tangent Taylor's Theorem tesimal true value vertex xn+1 xo+h Δα Δυ
Popular passages
Page 167 - A tangent to a circle is perpendicular to the radius drawn to the point of contact.
Page 93 - The angle formed by a tangent and a chord is measured by half the intercepted arc.
Page 190 - We can now take up some new problems that could not be conveniently approached while the integral was treated merely as an inverse function, and we shall consider very briefly one connected with the subject of centre of gravity. The centre of gravity of a body is a point so situated that the body will remain motionless in any position in which it may be placed, provided this point is supported.
Page iii - Its peculiarities are the rigorous use of the Doctrine of Limits, as a foundation of the subject, and as preliminary to the adoption of the more direct and practically convenient infinitesimal notation and nomenclature ; the early introduction of a few simple formulas and methods for integrating ; a rather elaborate treatment of the use of infinitesimals in pure geometry ; and the attempt to excite and keep up the interest of the student by bringing in throughout the whole book, and not merely at...
Page 11 - For any particular value of a;, this limit, as we shall see, will, in general, have a perfectly definite value ; but it will change in value as x changes ; that is, the derivative will, in general, be a new function of x.
Page 252 - FMa=f™a, the curves are said to have contact of the nth order at the point whose abscissa is a. • Contact of a higher order than the first is called osculation. 238. The difference between the ordinates of points of the two curves having the same abscissa and infinitely near the point of contact, is an infinitesimal of an order one higher than the order of contact of the curves. Let...
Page 164 - P'S = P'F, from the definition of a parabola ; .-. P'N=P'Q. P'N cosPP'S = pp.' COB pf= iimit cos/ ilmit cosT'PK P±P = P'™P FV =1' by Art. 162; .-. T'PR= T'PF, and the tangent at any point of a parabola bisects the angle between the focal radius and the diameter through the given point.
Page 168 - EXAMPLE. Prove that a tangent to an hyperbola bisects the angle between the focal radii drawn to the point of contact. 168. To find the area of a segment of a parabola cut off by a line perpendicular to the axis. Compare the required area with the area of the circumscribing rectangle. We can regard the...
Page 33 - Dxy, which is, by Art. 27, the tangent of the inclination of the curve to the axis, must equal zero. Of course it does not follow from the argument just presented, that every value of x that makes...