Calculating the Normal Equation to a Parabolic Curve at a Specific Point

The normal line to a curve at a specific point is a line that is perpendicular to the tangent line at that point. Understanding the process to find the equation of the normal line is essential in several areas of mathematics and physics. This article will walk you through the steps to determine the equation of the normal to the curve y x^2 - x^4 at the point (-1, 6).

Step 1: Finding the Derivative

To find the slope of the tangent line to the curve at any point, we first need to derive the function y x^2 - x^4. The derivative, denoted as y', will give us the slope of the tangent line at any point on the curve.

Given curve: y x^2 - x^4

The derivative of the function (y') is: 2x - 4x^3

Step 2: Evaluating the Derivative at the Given Point

Next, we evaluate the derivative at the given point, which is (-1, 6).

At x -1, y' 2(-1) - 4(-1)^3 -2 - 4(-1) -2 4 2

Thus, the slope of the tangent line at the point (-1, 6) is 2.

Step 3: Determining the Slope of the Normal Line

The slope of the normal line is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent line is 2, the slope of the normal line is:

-frac{1}{2}

Step 4: Equation of the Normal Line

We use the point-slope form of a line equation, which is:

y - y_1 m(x - x_1)

Here, the point (x_1, y_1) is (-1, 6) and the slope m is -frac{1}{2}. Plugging in these values, we get:

y - 6 -frac{1}{2}(x - (-1))

which simplifies to:

y - 6 -frac{1}{2}x - frac{1}{2}

Finally, rearranging the equation, we have:

y -frac{1}{2}x 6 - frac{1}{2}

y -frac{1}{2}x frac{11}{2}

Therefore, the equation of the normal to the curve y x^2 - x^4 at the point (-1, 6) is:

y -frac{1}{2}x frac{11}{2}

Final Answer

The final equation of the normal line is y -frac{1}{2}x frac{11}{2}.

Conclusion

Understanding how to derive the equation of the normal line to a curve at any specified point is a fundamental concept in calculus. This process involves several key steps: finding the derivative, evaluating it at the given point, determining the slope of the normal, and using the point-slope form to complete the equation. The example given above illustrates the application of these steps. Proficiency in these techniques is valuable for advanced mathematical and scientific problems.