An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

SINGULAR POINTS OF CURVES.

159

If f(x,) = positive, and f/(2) = negative, fig. 53 represents the curve.
(2) Let f(a) be a finite quantity; take MP =f(a),
fig. 54, and draw the line O'PO parallel to AM; then
f() -f(a) represents the distance of any point on the
curve above O'PO. Hence
If f' (a) = o, f(ix) -f(a) positive, f(x,)-f(a)= positive,
the tangent at P coincides with MP, but the curve lies above
OPO on each side of MP; therefore fig. 54 represents its
form.
If f'(a) = co, f(xi) -f (a) = negative, and f () - f (a)
=negative; then fig. 55 represents the curve.
If f'(a) = co, f(,) - f(a) = negative, and f(V)2) -f(a)
= positive; then the curve touches MP at P, lies below it
on the left side of MP, and above it on the right; therefore fig. 56 represents it.
If f'(a) = co, f(xi) -f(a) = positive, and f(x,) -f(a)
= negative then fig. 57 represents the curve.
If, however, f'(a) be a finite quantity, draw the line
T'PT (fig. 58) making an angle tan-'f'(a) with the axis of
o; then this line is the tangent to the curve at P, and, its
equation being y' -f(a) =f'(a) (xv - a),
y - y' or f(x) -f (a) - f'(a) (v - a)
is the distance of any point of the curve above this line.
Hence, if for brevity, we put
S (x) =f () -f(a) -f (a) (,v - a),
it is evident that:
If qp (i) = positive, and (p ()) = positive, the curve lies
above the tangent T'PT on both sides of MP, and is therefore represented by fig. 59.
If ) (x,) = negative, and p (x,) = positive, the curve lies
below T'PT on the left side of MP, and above it on the
right, and is therefore represented by fig. 60.