# Applications of Derivatives, Equation of Tangent and Normal to a Curve, Mathematical Formulation of Rolle's Theorem

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## Equation of Tangent and Normal to a Curve

We know that the equation of a line passing through a point and with slope is

As discussed in the section before, the slope of tangent to the curve at is given by at and that of normal is at

Equation of tangent to the curve at the point is

And, the equation of normal to the curve at the point is

Example:

Find the equation of the tangent and normal to the circle at the point

Solution:

The equation of circle is

Differentiating , w.r.t. x, we get

Equation of tangent to the circle at is

or or

Also, slope of the normal

Equation of normal to the circle at is

or

Equation of tangent to the circle at is

Equation of normal to the circle at is

## Mathematical Formulation of Rolle’S Theorem:

Let f be a real function defined in the closed interval such that

(i) is continuous in the closed interval

(ii) is differentiable in the open interval

(iii)

Then there is at least one point c in the open interval such that

Example:

Verify Rolle’s for the function

Solution:

(i) is a polynomial function and hence continuous in

(ii) is differentiable on

(iii) Also and

All the conditions of Rolle’s Theorem are satisfied.

Also,

gives

We see that both the values of lie in

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