The Drawing of a Tangent to a Curve

Excerpts

• The Drawing of a Tangent to a Curve

  The line $TPV$ indicates the direction in which the point $P$ is proceeding, and is called the *tangent* of the curve at the point $P$.

• The Drawing of a Tangent to a Curve

  If, therefore, a line $PV$ be drawn through $P$, making with $PQ$ an angle whose tangent is $2x$, the chord $PP'$ will, as $P'$ approaches towards $P$, or as $dx$ is diminished, continually approximate towards $PV$, so that the angle $P'PV$ may be made smaller than any given angle, by sufficiently diminishing $dx$.

• The Drawing of a Tangent to a Curve

  Since the relation $y = x^{2}$ is true for the co-ordinates of every point in the curve, we have $y + dy = (x + dx)^{2}$, the subtraction of the former equation from which gives $dy = 2x\, dx + (dx)^{2}$, or $\dfrac{dy}{dx} = 2x + dx$.

• The Drawing of a Tangent to a Curve

  There is some confusion between these different uses of the word tangent.

• The Drawing of a Tangent to a Curve

  If the curve were the interior of a small solid tube, in which an atom of matter were made to move, being projected into it at $O$, and if all the tube above $P$ were removed, the line $PV$ is in the direction which the atom would take on emerging at $P$, and is the line which it would describe.

• The Drawing of a Tangent to a Curve

  This problem, of drawing a tangent to any curve, was one, the consideration of which gave rise to the methods of the Differential Calculus.

Equations

• This equation is in Algebraical Geometry (Algebraical Geometry)

• The Drawing of a Tangent to a Curve

  y + dy = (x + dx)^{2}

  When x increases by dx, the point on the curve has co-ordinates x + dx and y + dy, and the curve relation still holds for them.

• This equation is in The Notation of the Differential Calculus (The Notation of the Differential Calculus)
• This equation is in The Notation of the Differential Calculus (The Notation of the Differential Calculus)

• The Drawing of a Tangent to a Curve

  y - dy = (x - dx)^{2}

  When P' is placed on the other side of P, at co-ordinates x - dx and y - dy, the curve relation holds for those co-ordinates.

• The Drawing of a Tangent to a Curve

  dy = 2x\, dx - (dx)^{2}

  Subtracting the backward relation from y = x^2 gives the increment of y for the backward step.

• The Drawing of a Tangent to a Curve

  \dfrac{dy}{dx} = 2x - dx

  For the backward step, the ratio of the increment of y to the increment of x equals 2x minus dx, which approaches 2x as dx diminishes.

• The Drawing of a Tangent to a Curve

  y = \phi x

  The ordinate y is some function of the abscissa x, written as phi x.

• The Drawing of a Tangent to a Curve

  y = \log x

  For the curve whose ordinates are the Naperian logarithms of the abscissae, y equals the natural logarithm of x.

• The Drawing of a Tangent to a Curve

  y + dy = \log x + \dfrac{1}{x}\, dx - \dfrac{1}{2x^{2}}\, dx^{2}

  The incremented ordinate of the logarithmic curve, expanded in powers of dx, is log x plus dx/x minus dx^2/(2x^2), with further terms omitted (etc.).

Problems

No exercises in this chapter.