What are the Equations of the Tangent and Normal Lines to the Curve?

Finding the equations of the tangent and normal lines to a given curve involves a series of straightforward steps. In this article, we will explore the process using the curve y 3x2 - 2x 1 at the point (1, 2). By the end of this guide, you will understand the concepts of derivatives, point-slope form, and the geometric relationships between tangent and normal lines.

Step-by-Step Guide

Step 1: Calculate the Derivative

First, we need to find the derivative of the given curve, which gives us the slope of the tangent line at any point on the curve.

The curve is defined by the equation:

y 3x2 - 2x 1

To find the derivative, we differentiate the equation with respect to x using the power rule:

y' 6x - 2

Step 2: Evaluate the Derivative at x 1

Next, we substitute x 1 into the derivative to find the specific slope of the tangent line at the point (1, 2).

Substitute x 1 into the derivative:

y' 6(1) - 2 4

This value, 4, represents the slope of the tangent line at point (1, 2).

Step 3: Equation of the Tangent Line

Using the point-slope form of a line, we can write the equation of the tangent line. The point-slope form is given by:

y - y1 m(x - x1)

We know the slope m 4 and the point (1, 2), so we can write:

y - 2 4(x - 1)

By simplifying this equation, we get the equation of the tangent line:

y 4x - 2

Step 4: Slope of the Normal Line

The slope of the normal line is the negative reciprocal of the slope of the tangent line. In this case, the slope of the normal line is:

mnormal -1/4

Step 5: Equation of the Normal Line

Again, using the point-slope form, we can write the equation of the normal line:

y - 2 -1/4(x - 1)

By simplifying this equation, we get the equation of the normal line:

y -1/4x 9/4

Conclusion

In summary, the equations of the tangent and normal lines to the curve y 3x2 - 2x 1 at the point (1, 2) are:

Tangent Line: y 4x - 2 Normal Line: y -1/4x 9/4 The process of finding these lines involves taking the derivative of the curve to determine the slope at a specific point, and then using the point-slope form to write the equations of both the tangent and normal lines.