TANGENT AT A GIVEN POINT ON OTHER CURVES
The
technique used to find the slope and equation of the tangent line for a
standard parabola can be used to find the slope and equation of the tangent
line to a curve at any point regardless of the type of curve. The method can be
used to find these relationships for circles, hyperbolas, ellipses, and
general algebraic curves.
This
general method is outlined as follows: To find
the slope, m, of a given curve at the point
, choose a
second point, P', on the curve so that it has coordinates
; then
substitute each of the coordinates of P' and P_{1} in the equation of
the curve and simplify. Divide both
sides by
x and
eliminate terms that contain powers of
y higher
than the first power, as
previously discussed. Solve for
. Let
x approach
zero and
will approach the slope of the tangent line,
m, at point P_{1}.
When
the slope and coordinates of a point on the curve are known, you can find the
equation of the tangent line by using the pointslope method.
EXAMPLE: Using the method outlined, find the slope and equation of the tangent
line to the curve
SOLUTION: Choose a second point such that it has coordinates
Substitute
into equation (1)
Thus
Then
Divide
both sides by
x
^{}
and
eliminating (
y)^{2}
results in
Solve
for
Let
x
approach zero, so that
Now
using the pointslope form of a straight line, substitute
for m:
Multiply
both sides by y_{1}
Rearrange:
but
Then, by substitution
and
which is the general equation of the tangent line to the
curve
_{
}
EXAMPLE: Using the given method, with minor changes, find
the slope and equation of the tangent line to the curve
SOLUTION: Choose a second point such that
it has coordinates
Substitute into equation (1):
Since
, then
Then divide by
x and
eliminate (
y)^{2}
Solve for
Let Ax approach zero, so that
which is the slope desired.
Use the pointslope form of a straight line to find the
equation of the tangent line to the curve at point (x_{1},y_{1})
as shown in the following:
Substitute
for m:
Multiply both sides by y_{1}:
Rearrange to obtain
Substitute
;
Divide both sides by y_{1} to obtain
which is the equation desired.
PRACTICE PROBLEMS:
Find the slope and equation of the tangent line to the curve,
in problems 1 through 6, at the given points.
ANSWERS:
