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EQUATIONS OF TANGENT LINES AND NORMAL LINESSuppose we have a point 11,xyand a slope mof a line. Then the equation of the line through the point with that slope is11y ym x x  .You should remember this formula from algebra. It’s the point-slopeform of the equation of a line. If you don’t remember this formula, memorize it!Next, suppose that we have an equation yf x, where 11,xysatisfies that equation. Then, f xm, and we can plug all our values into the equation for a line and get the equation of the tangent line.Example 1Find the equation of the tangent line to the curve 25yxat the point 3, 45.Example 2Find the equation of the tangent line to the curve 32yxxat 3, 36Naturally, there are a couple of things that can be done to make the problems harder. First, you can be given only the x-coordinate. Second, the equation can be more difficult to differentiate.In order to find the y-coordinate, all you have to do is plug in the x value into the equation for the curve and solve for y.Example 3Find the equation of the tangent line to 2253xyxat 1x.
Sometimes, instead of finding the equation of a tangent line, you will be asked to find the equation of the normal line. A normal lineis simply the line perpendicular to the tangent line at the same point. You follow the same steps as with the tangent line, but you use the slope that will give you a perpendicular line. Remember what that is? It’s the negative reciprocal of the slope of the tangent line.Example 4Find the equation of the normal to 541yxx  at 2x.Example 5Find the equations of the tangent and normal lines to the graph of 2101xyxat the point 2, 4.Example 6The curve 2yaxbx c  passes through the point 2, 4and is tangent to the line 1yxat the point 0,1. Find the values of a, b, and c.Example 7Find the points on the curve 32231220yxxx   where the tangentis parallel to the x-axis.