Lecture 17 MATH 121 (Wg Cdr Monir)

 MATH 121 (Wg Cdr Monir) 

Tangent and Normal

1. The tangent at P to a given curve is defined as the limiting position of the secant PQ (when such a limit exists) as the point Q approaches P along the curve.

    Derivative of any point on a curve is the slope of the tangent at the point (x,y) on the curve.
That is $\frac{dy}{dx} = \tan \psi =$ slope of the tangent.
In a curve $y = f(x)$, if at a particular point (x,y)
    1.  $\frac{dy}{dx} = 1$, Then the slope of the tangent at that point of the curve makes an angle of $45^\circ$ with the positive direction of x-axis.
    2. $\frac{dy}{dx}$ = Negative, Then the tangent to the curve at that point makes an obtuse angle with the positive direction of x-axis.
    3.  $\frac{dy}{dx} = 0$ , Then the tangent to the curve at that point makes an angle of $0^\circ$ with the x-axis, i.e the tangent to the curve at that point is parallel to the x-axis.
    4. $\frac{dy}{dx} = \infty$, then the tangent to the curve at that point makes on angle of $\frac{\pi}{2}$ with the x-axis, i.e the tangent to the curve at the point is perpendicular to x-axis.

    Equation of tangent at any point (x,y) of the curve  $y = f(x)$
  $$Y - y = \frac{dy}{dx} (X - x)$$  

    Equation of the tangent at any point on the curve $f(x,y) = 0$  $$(X - x) f_x + (Y - y) f_y = 0$$  

If the curve be given by the algebraic equation  $f(x,y) = 0$ which can be made homoheneous in x,y and z the equation of the tangent will be  $X f_x + Y f_y + Z f_x = 0$  

    Equation of the tangent at any point on the curve $f(x,y) = 0$
if the equation of the curve be given in the form  $f(x,y) = 0$ then  $\frac{dy}{dx} = \frac{ - \frac{\partial f}{\partial x}}{\frac{\partial f }{\partial y}}$  
substituting this expression for $\frac{dy}{dx}$ in  $$\begin{align*} & Y - y = \frac{dy}{dx} (X - x)\\ \textrm{we have} &, \\ & Y - y = \frac{ - \frac{\partial f}{\partial x}}{\frac{\partial f }{\partial y}} (X - x)\\ or, ~ & (X - x) \frac{\partial f}{\partial x} + (Y - y) \frac{ \partial f}{\partial y} = 0 \end{align*}$$  
as the equation of the tangent.
Replacing $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ by $f_x$ and $f_y$ respectively we may write the equation of the tangent in the form  $(X - x) f_x + (Y - y) f_y = 0$  

    ♦ Normal and its equation.
Normal: Normal to a curve at a point is defined as the straight line through that point at right angles to the tangent to the curve at that point.
We know that tangent to a curve  $y = f(x) ~ \textrm{at} ~ (x,y) ~\textrm{is}$  $Y - y = \frac{dy}{dx} (X - x) \tag{1}\label{eq:1}$  
any line throught(x,y) is given by  $Y - y = m (X - x) \tag{2}\label{eq:2}$
If this line is normal to the curve at (x,y) this is perpendicular to the tangent to the curve, i.e 

  $$\begin{align*} & m \frac{dy}{dx} = - 1\\ \therefore ~& m = - \frac{1}{\frac{dy}{dx}} \end{align*}$$  
From (2) the equation of the normal at (x,) to the curve  $y = f(x)$ is  $$\begin{align*} & Y - y = - \frac{1}{\frac{dy}{dx}} (X - x) \\ or, ~ & (Y - y ) \frac{dy}{dx} + (X - x) = 0 \end{align*}$$