Elements of the Differential Calculus: With Examples and Applications |
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Common terms and phrases
Analytic Geometry angle approaches zero axis Binomial Theorem body called centre chord circle constant coördinates corresponding increments curvature cycloid D₁s D₂ D₂u D₂y decreasing definite value derivative descent difference differential earth ellipse equal zero equation evolute EXAMPLES expression finite formulas function fx Fx fx-fa Geometry given curve given point hence higher order hn+1 increases indefinitely independent variable Indeterminate Forms infinitely near points integral length limit Ay limiting position locus loga logx Maxima and Minima maximum mean velocity minimum values multiplied negative normal obtained ordinate parabola perpendicular plane principal infinitesimal problem quantity radii radius radius of curvature ratio rectangle secant line second order secx sin² sinx Suppose tangent Taylor's Theorem tesimal true value vertex vertical xn+1 Δα Δη Δυ
Popular passages
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 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'F, from the definition of a parabola ; PB = PF=QF, .-. P'N=P'Q. P'M cosPP'F= pp/' P' V cosPP'AS= ^ cosT'PF^ iimit rcps/Wl limit rP^TI cosT'PR P=P[_coaPP'Sj P
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 167 - If two right-angled triangles have the hypothenuse and a side of the one equal to the hypothenuse and a side of the other, each to each, the triangles are equal. Let...
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...
Page 81 - As the curvature of a circle has been found to be the reciprocal of its radius, a circle may be drawn which shall have any curvature required. A circle tangent to a curve at any point, and having the same curvature as the curve at that point, is called the osculating circle of the curve at the point in question. Its...
Page 11 - ... increment of the independent variable approaches zero. Such a limit is called a derivative, or a differential coefficient, and the study of its form and properties is the fundamental object of the Differential Calculus. CHAPTER II. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 18. If y is a function of x...