Calculus: Analytic Geometry and Calculus, with Vectors

346 Functions, graphs, and numbers denote, respectively, the least upper bound of all lower Darboux sums and the greatest lower ...

5.7 Darboux sums and Riemann integrals 347 All this gives the following theorem which involves the numbers L and U defined in th ...

348 Functions, graphs, and numbers Similarly, we apply the fundamental Theorem 5.731 to prove Riemann integrability of continuou ...

5.7 Darboux sums and Riemann integrals 349 whenever xk-1 < xk < xk for each k. Defining Mk and mk by (5.712), we conclude ...

350 Functions, graphs, and numbers represents an Archimedes integral, then the integral exists and has the value 1, but that if ...

5.7 Darboux sums and Riemann integrals 351 ThusF'(x) exists when -1 =< x 5 1. As x approaches 0, F(x) oscillates between -1 a ...

352 Functions, graphs, and numbers g(x) > f(x) for each x for which x > xo. Progress with the theory depends upon the basi ...

5.7 Darboux sums and Riemann integrals 353 the interval a<= t =< x. Prove that T(x), P(x), N(x) are all nonnegative and mo ...

6 Cones and conics 6.1 Parabolas Before plunging into the general aspects of this chapter, we obtain more information about the ...

6.1 Parabolas 355 point must therefore satisfy the equation of the parabola, so? = k(2 ?) 2 and ? = 1/4k. The coordinates of F a ...

356 Cones and tonics a line on the cone. The intersection is a curve of which a part (the solid part) lies on the front half of ...

6.1 Parabolas 357 These matters are important because equations of the form (2) appear very often, but for basic studies of para ...

Cones and conics 8 Let P, be a point on a parabola which is not the vertex P. Let W be the 358 intersection of the tangents to t ...

6.1 Parabolas 359 12 This problem and Figure 6.193 delve a bit deeper into the geometry of parabolas. The figure shows the pa- r ...

360 Cones and conics that the two points Pi(xi,kxi), P2(xs,kz2) on the graph of y = kx2 are end points of a focal chord if and o ...

and hence if (2) If (2) holds, then (3) and hence either (4) or (5) 6.2 Geometry of cones and conics 361 Vx2 +y2= a ± y. x2 + y2 ...

362 Cones and conics in this chapter lies in conics of less simple natures. These turn out to be parabolas and ellipses, each of ...

6.2 Geometry of cones and conics 363 because the vector DP makes the angle S with vertical lines. Equating the two expressionsfo ...

364 Cones and conis P(x y) know the eccentricity e, the coordinates (x1jy1) of a locus F, and the equation F(x,,y,) 'IX + By + C ...

6.2 Geometry of cones and conics 365 Putting this value of xl in (6.261) gives the equation (6.271) (1 - e2)x2 + y2 = e2p2 -e2 T ...