Tangents and Normals
A tangent or normal to a curve is a line, taking the form
where
is the gradient and
is the intercept. Given a function
we can find the gradient at
by finding the gradient function
and substituting the valueinto this expression. Sometimes however we don't have
so
is not given explicitly as a function of
In these cases typically we have to differentiate implicitly and findas a function of both
and
and then substitute a point
into the expression forto find the gradient at that point. Finally substitute into the equation
to find the equation of the line.
Example: Find the equation of the tangent to the curve
at the point
We differentiate implicitly to get
The gradient at the point
is
Example: Find the equation of the tangent to the curve
at the point
We differentiate implicitly to get
We have to make
the subject of this equation.
The gradient at the point
is
Example: Find the equation of the normal to the curve
at the point
We differentiate implicitly to get
We have to makethe subject of this equation.
The gradient at the point
is
