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 P₁ 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 P₁.

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 point-slope 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 point-slope form of a straight line, substitute for m:

Multiply both sides by y₁

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 point-slope form of a straight line to find the equation of the tangent line to the curve at point (x₁,y₁) as shown in the following:

Substitute for m:

Multiply both sides by y₁:

Rearrange to obtain

Substitute ;

Divide both sides by y₁ 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.

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