Finding the Equation of the Tangent Line to a Curve

In this article, we will learn how to find the equation of the tangent line to the curve y = x^3 - 3x + 1 at the point (1, -1).

Step 1: Understand the Concept of a Tangent Line

A tangent line is a line that touches a curve at a single point, and it is perpendicular to the curve's slope at that point. The equation of a tangent line can be expressed in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

Step 2: Find the Derivative of the Curve

To find the slope of the tangent line, we need to find the derivative of the curve at the given point. The derivative of y = x^3 - 3x + 1 is y' = 3x^2 - 3.

Step 3: Evaluate the Derivative at the Given Point

Substituting x = 1 into the derivative expression, we get: y' = 3(1)^2 - 3 = 0 - 3 = -3

Step 4: Find the Slope of the Tangent Line

The slope of the tangent line is equal to the derivative of the curve at the given point. Therefore, the slope of the tangent line is -3.

Step 5: Find the y-Intercept of the Tangent Line

To find the y-intercept of the tangent line, we can use the point-slope form of the equation: y - y1 = m(x - x1) Substituting the known values, we get: y - (-1) = -3(x - 1) Simplifying, we get: y + 1 = -3x + 3 Rearranging, we get: y = -3x + 4

Conclusion

The equation of the tangent line to the curve y = x^3 - 3x + 1 at the point (1, -1) is y = -3x + 4.