An elementary treatise on the differential and integral calculus, with numerous examples

Edward Albert Bowser

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1886 Excerpt: ...of the tangent. Since the normal is perpendicular to the tangent at the point of tangency, its equation is, from (2), Eem.--To apply (2) or (3) to any particular curve, we substitute for----, or-=--,, its value obtained from the equation of the curve and expressed in terms of the co-ordinates of the point of tongency. EXAMPLES. 1. Find the equations of the tangent and normal to the ellipse ay + bh = aW. We find dJL=-x.... l=_TM. dx a2y' dx' ah/ and this value in (2) gives, y-y =-aYx-ar; which by reduction becomes, tfyy' + Wxx'--aW, which is the equation of the tangent; and y-y =dfx-x) is the equation of the normal. 2. Find the equations of the tangent and normal to the parabola y2 = 2px. We find f=P-,.:% =?-„ dx y dx y and this value in (2) gives y-y =5 (-), or yy'--y'2 = px--px'. But y' = 2px'; 101. Length of Tangent, Normal, Subtangent, Subnormal, and Perpendicular on the Tangent from the Origin. Let PT represent the tangent at the point P, PN the normal; draw the ordinate PM; then Subtangent--y the perpendicular from origin on tqngent _ aW " (a4y" + 6V2)' 2. Find the subtangent and subnormal to the Cissoid 3 Here 3. Find the value of the subtangent of y2 = 3x2--12, at x = 4. Subtangent = 3. 4. Find the length of the tangent to f = 2x, at x = 8. Tangent = 4/l7. 5. Find the values of the normal and subnormal to the cycloid (Anal. Geom., Art. 156). dx V2ry--y.'. Subnormal = V%ry--y2 = MO. Normal = V2ry--PO. It can be easily seen that PO is normal to the cycloid at P; for the motion of each point on the generating circle at the instant is one of rotation about the point of contact O, i.e., each point for an instant is describing an infinitely small circular arc whose centre is at O; and hence PO is normal to the curve, i.e., the normal passes...

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ISBN 10:  1236951565 ISBN 13:  9781236951564
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