O
Level Additional Maths Notes: 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 make
the
subject of this equation.
The gradient at the point
is
