. Differential and integral calculus, an introductory course for colleges and engineering schools. 136 DIFFERENTIAL CALCULUS this branch is symmetrical about the initial line OA. By plotting a fewpoints in the angle BOA, the curve can be drawn with considerableaccuracy. The other branch p= — Vcos 2 0 is of the same size and shape and lies in the angle BOC. The curvehas a double point at the pole and eachbranch has a flex there. The curve has now been drawn with-out any aid from Calculus. Some ofthe foregoing conclusions, however, arenot quite warranted. For example, weare not yet quite sure th
. Differential and integral calculus, an introductory course for colleges and engineering schools. 136 DIFFERENTIAL CALCULUS this branch is symmetrical about the initial line OA. By plotting a fewpoints in the angle BOA, the curve can be drawn with considerableaccuracy. The other branch p= — Vcos 2 0 is of the same size and shape and lies in the angle BOC. The curvehas a double point at the pole and eachbranch has a flex there. The curve has now been drawn with-out any aid from Calculus. Some ofthe foregoing conclusions, however, arenot quite warranted. For example, weare not yet quite sure that the lines BBand CC are actually tangents, and we arenot at all sure that the curve is roundedat the points A and A, as shown in the figure on page 135. It may havecusps at these points, with the line AA for cuspidal tangents, as in theaccompanying figure. To settle these doubts, we make use of formula. (a). Differentiating p- aVcos 2 0, we haven a sin 2d V cos 2 0 whence tan = ■—- = — cot 2 0, and therefore 4> = =b 2 0. When 0 = ± -, = ir or 0, and this means that the tangent at the pole coincides with and has the same direction as the radius vector there, that is, that the lines BB and CC are tangents at the pole. Since the curve crosses its tangent at 0, this point is a flex on each branch. When q0 = 0, = -, and when 0 = ir, = -~ , which means that the tangents at A and A are perpendicular to AA. Hence the curve is rounded at Aand A as shown in the first figure. From the relation = ^ + 20 it follows that, to draw the tangent to the curve at a given point of it, it is only necessary to let fall from thatpoint a perpendicular upon a line through 0 which makes an angle 3 0with AA. It has not been shown that thecurve may not have undulationsas in the accompanying figure. Arigorous proof that such is not the case we do not giv
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
