By A. C. Burdette (Auth.)
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Extra resources for An Introduction to Analytic Geometry and Calculus
001, n n > 1000. Similarly, it is clear that t We are going to be interested solely in infinite sequences, so we state our definition accordingly. 4-4. LIMIT OF A SEQUENCE 55 In general, if we require sn to be less than any previously assigned positive number, say ε, we have 1 sn = - < ε, n 1 η > -. ε A very important point to note here is that it is possible to find not only a term in the sequence that is less than ε, but also that every succeeding term in the sequence is less than ε. 01, but so is every sn with n > 100.
However, the parabola 36 3. NONLINEAR EQUATIONS AND GRAPHS would be reversed and the vertex would become the right-most point on the curve. If we combine constants,t (3-5) takes the form k(x -t) = (y- s)2 (3-6) and we may summarize the preceding observations as follows. THEOREM 3-4. An equation reducible to the form (3-6) is the equation of a parabola having the following properties : (a) Its vertex is at (t, s). (b) If k is positive (negative) it extends to the right (left) from the vertex. (c) It is symmetric to its axis y = s.
We have \PF\ = y/(x - r)2 + (y- s)2. Since the coordinates of P satisfy (3-5) we may write PF\= (x-ry = Jx2-2xp + 2(r-p) (-Ψ) + p2 = \x-p\ = \PQl as was to be shown. Hence we may state that (3-5) is an equation of the parabola described. It is clear from the definition that the point V, midway between the focus and directrix, is a point on the parabola; moreover, under the assumption that r > p, it is the left-most point on the curve. Its coordinates, from (1-4) and (1-5), are [(r + p)/2, s]. This point, whose coordinates may be read directly from (3-5), is called the vertex of the parabola.