. Differential and integral calculus, an introductory course for colleges and engineering schools. t is not actually necessary to note the points of maximum andminimum coordinates: they appear of themselves as the severalarcs are drawn. Although y changes sign at t = -ji=, this is not a point of max- imum or minimum y, and although y changes sign at t 1</2 and at t = oo, the curve has no flexes whatever. Let the studentexplain this. It may be observed in passing that in this example y is a three-valued function of x. Let the student calculate the values thatx and y have when t = 1 and — |(1
. Differential and integral calculus, an introductory course for colleges and engineering schools. t is not actually necessary to note the points of maximum andminimum coordinates: they appear of themselves as the severalarcs are drawn. Although y changes sign at t = -ji=, this is not a point of max- imum or minimum y, and although y changes sign at t 1</2 and at t = oo, the curve has no flexes whatever. Let the studentexplain this. It may be observed in passing that in this example y is a three-valued function of x. Let the student calculate the values thatx and y have when t = 1 and — |(1 db V5). 90. Geometric Interpretation of the Parameter. When thecurve is rational and the relation y = tx holds, as in the case ofthe folium, the parameter, t, admits of a simple and interestinggeometrical interpretation. The equation y = tx represents aline through the origin with the slope t. Let us find the intersec-tions of this line with the folium W 3 axy + x3 = 0. Eliminating y from these equations, wehave x2(t*x- Sat + x) = 0, whence x2 = 0 or x = and y = 0 or y = Sat1 + t3Sat21 + t3. The solution x2 = 0, y = 0, means that the line y = tx meets thecurve twice at the origin, which means that the origin is a doublepoint. The other solution gives the coordinates of P, the onlyintersection outside the origin. As t varies, the line y = txturns about the origin as a pivot, while the variable point Ptraverses the line and traces the curve. And in expressing the 124 DIFFERENTIAL CALCULUS §91 x- and ^/-coordinates of this tracing point, P, in terms of thevariable slope t, we have the parametric equations of the curve. 91. Exercises. Draw the following curves. In every casederive the x- and ^/-equation from the parametric equations. Manyof these curves have already been given by their x- and ^/-equationsin preceding exercises. (The figures stand above their equations.) 1. x=10(£2-l), y=tx. 2. x=t2+2, y= the values of x and y when t=i V2,
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