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: