. An elementary treatise on the differential and integral calculus. axis of x, y is therefore, y -v4 is — ; that is, y d2y dx2 :~ > - * dx2 is — when the curve is concave towards the axis of x. In d*y the same way it may be shown thatcurve is convex towards the axis of x. dx2 is -f, when the 107. Polar Co-ordinates.—A curve referred to polarco-ordinates is said to be concave or convex to the pole atany point, according as the curve in the neighborhood ofthat point does or does not lie on the same side of the tan-gent as the pole. EXAMPLES. 193 It is evident from Fig. 24, that when the curve
. An elementary treatise on the differential and integral calculus. axis of x, y is therefore, y -v4 is — ; that is, y d2y dx2 :~ > - * dx2 is — when the curve is concave towards the axis of x. In d*y the same way it may be shown thatcurve is convex towards the axis of x. dx2 is -f, when the 107. Polar Co-ordinates.—A curve referred to polarco-ordinates is said to be concave or convex to the pole atany point, according as the curve in the neighborhood ofthat point does or does not lie on the same side of the tan-gent as the pole. EXAMPLES. 193 It is evident from Fig. 24, that when the curve is con-cave toward the pole 0, as r increases p increases also, and drtherefore -j- is positive ; and if the curve is convex toward the pole, as r increases^ decreases, and therefore -=- is negative. If thereforedp the equation of the curve is given in terms of r and 6, to find whether the curve is concave or convex towards the pole, we must transform the equation into its equivalent between r and p, by means of (10) in Art. 102, and then find ^- Fig. EXAMPLES. 1. Find the direction of curvature of Here _ (*? ~ 1) (x - 3) y — x — 2 dx2 ~ 2 (x - 2)3 (Py that is, -t4 is positive or negative, according as z2; and therefore the curve is convex downward for all valuesof x 2. 2. Find the direction of curvature of y = I? -\- c {x + a)2 and y = a2 Vx — a. A?is. The first is concave upward, the second is concavetowards the axis of x. 3. Find the direction of curvature of the lituus r = —-t> 0* 194 SINGULAR POINTS. dr r r* #Mere dd~ ~ 26~ ~%a^ W - la* which in (10) of Art. 102 gives, 2aV a?r (4a4 + r4)^ P (r4 + ^4)1 ^ ~ 2«2 (4«* -r4) Therefore the curve is concave toward the pole for valuesof r a V%- 4. Find the direction of curvature of the logarithmicspiral r = a9. By Art. 102, Ex. 2, mr dr ^/m% + 1 V Vm* + 1 dP m which is always positive, and therefore the curve is alwaysconcave toward the pole. SINGULAR POINTS. 108. Singula
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