Quantcast Tangent at a Given Point on Other Curves

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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.

ANSWERS:




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