Understanding the Normal Line to a Curve

When given a curve and a specific point on that curve, the process of finding the normal line involves several key steps. The normal line is a line that is perpendicular to the tangent line at the given point. This article will guide you through the process of finding the normal line, providing step-by-step instructions and examples.

Step-by-Step Guide

Find the Slope of the Tangent Line

The first step is to determine the slope of the tangent line at the given point. This is achieved by differentiating the equation of the curve to find the derivative dy/dx, which represents the slope of the tangent line at any point on the curve.

Substitute the Given Point into the Derivative

Once you have the derivative, substitute the coordinates of the given point into the derivative to find the slope at that specific point. This slope is denoted as m.

Determine the Slope of the Normal Line

The slope of the normal line is the negative reciprocal of the slope of the tangent line. If the slope of the tangent line is m, then the slope of the normal line is -1/m.

Form the Equation of the Normal Line

Using the slope of the normal line and the given point, you can write the equation of the normal line in point-slope form, which is:

y - y? m_normal(x - x?)

Where m_normal is the slope of the normal line, and (x?, y?) are the coordinates of the given point.

Example Problem

Consider a curve given by y2 5x and a point on this curve (25, 5).

Differentiate the Given Curve

Differentiate the equation y2 5x to find the derivative:

2yy' 5

Solving for y':

y' 5 / (2y)

Evaluate the Derivative at the Given Point

Substitute the point (25, 5) into the derivative y' 5 / (2y) to find the slope of the tangent line at this point:

y' 5 / (2 * 5) 1/2

Find the Slope of the Normal Line

The slope of the normal line is the negative reciprocal of the slope of the tangent:

m_normal -1 / (1/2) -2

Write the Equation of the Normal Line

Using the slope of the normal line -2 and the given point (25, 5), the equation of the normal line is:

y - 5 -2(x - 25)

Expanding and simplifying:

y - 5 -2x 50

y 2x 55

Therefore, the equation of the normal line is:

2x y 55

Further Steps and Examples

For further understanding, consider the following additional steps and examples:

Carefully differentiate the given function

Find the derivative of the function y mx b and evaluate it at the specific point.

Determine the slope of the tangent line

Find the slope of the tangent line by evaluating the derivative at the given point.

Find the slope of the normal line

The slope of the normal line is the negative reciprocal of the slope of the tangent line.

Form the equation of the normal line

Use the slope of the normal line and the given point to write the equation of the normal line.

Conclusion

By following these detailed steps and examples, you can efficiently find the equation of the normal line to a curve at a given point. Understanding the process is essential for solving similar problems in calculus and applied mathematics.