Understanding Tangent Lines to Curves: A Step-by-Step Guide

In calculus, finding the equation of a tangent line to a curve at a given point is a fundamental concept. This article will guide you through the process, using the function y x^2 - 2x 1, and demonstrate how to find the tangent line at the point (0, 1).

1. The Derivative and Slope of the Tangent Line

The derivative of a function at a particular point gives us the slope of the tangent line at that point. For the curve y x^2 - 2x 1, we first need to find the derivative.

1.1 Deriving the Function

The given function is:

y x^2 - 2x 1

To find the derivative of this function, we differentiate with respect to x:

y d y d x 2 x - 2

The derivative, y', is 2x - 2.

2. Finding the Slope at the Given Point

We need to find the slope of the tangent line at the point (0, 1). We substitute x 0 into the derivative:

Slope at (0, 1): y' 2(0) - 2 -2

3. Using the Point-Slope Form

The point-slope form of the equation of a line is given by:

y - y1 m(x - x1)

where m is the slope, and (x1, y1) is a point on the line. In this case:

x1 0 y1 1 m -2

3.1 Constructing the Equation

Substituting the values into the point-slope form:

y - y 1 1 m ? x - x 1 y - 1 m ? x - 0 y - 1 m ? x y - 1 - 2 ? x y - 1 - 2 x y - 2 x - 1

Therefore, the equation of the tangent line to the curve y x^2 - 2x 1 at the point (0, 1) is:

y -2x - 1

4. Additional Insights

If you substitute x 0 into the derivative, you get m -2, the slope of the tangent line. Since the tangent line passes through the point (0, 1), the y-intercept (c) is 1. Using the slope-intercept form (y mx c), we can write:

y -2x - 1

5. Conclusion

In this article, we have demonstrated how to find the equation of a tangent line to the curve y x^2 - 2x 1 at the point (0, 1). The process involves finding the derivative, determining the slope at the given point, and substituting the values into the point-slope form or slope-intercept form. Understanding these steps and their applications can help you solve similar problems more efficiently.