Calculus How To

Normal Line

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In calculus, “Normal” has a meaning that is not immediately intuitive.  Normal within this context means perpendicular.  Therefore, a normal line to a point on a curve is the line that runs through that point and is perpendicular to the tangent line.

The following image shows the normal line running through the curve and perpendicular to the tangent:
normal line


Since the normal line and tangent line are perpendicular their slopes are opposite reciprocals. You can use the slope of the tangent line to find the slope of the normal line to the curve. 

There is an important rule that you must keep in mind:
Where two lines are at right angles (perpendicular) to each other, the product of their slopes (m1∙m2) must equal -1. 
So, once you have the slope of the tangent line you can figure out the slope of the normal line by making sure the product of the two slopes equals -1.  Once you have the slope of the normal line you can work out the equation of the normal to the plotted curve.

Example: Finding the Equation for the Normal Line

Find the equation of the normal line to the curve y = x3 at the point (2, 8).

Part One: Calculate the Slope of the Tangent

When working with a curve on a graph you must find the derivative of the function which gives us the slope of the tangent line.

  • y = x3
  • y′ = 3x2

The slope of the tangent when x = 2 is 3(2)2 = 12

The question may ask you for the equation of the tangent in addition to the equation of the normal line.  Since you already have the slope of the tangent the equation is relatively easy to find, using the formula for a linear equation (y = 12x – 16).

Part Two: Calculate the Slope of the Normal Line

The rule from above stated that the products of the two slopes must equal – 1.  Since you know the slope of the tangent is 12 (from part one), you need a number that multiplied by 12 equals -1.  That number is -1/12:

  • m1∙m2 = 12(-1/12) = -1
  • The slope of the normal line is -1/12.

Part Three: Find the Equation of the Normal


Once you have the slope of the normal line, the equation is, again. relatively easy to find.  The equation at the point (2, 8) is:


  • y – 8 = -1/12(x-2)
  • 12y + x = 98
CITE THIS AS:
Stephanie Glen. "Normal Line" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/normal-line/
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