Question

Find an equation of the tangent line to the curve y = x² - 2x at the point (1, 1/3).

Answer 1

The equation of the tangent line to the curve y = x² - 2x at the point (1, -1) is y = -1. This is found by taking the derivative of the function to find the slope, which is 0, and using the point-slope form with the point (1, -1).

Step-by-step explanation:

To find an equation of the tangent line to the curve y = x² - 2x at the point (1, -1), we need to follow these steps:

1. Determine the derivative of the function y = x² - 2x to find the slope of the tangent line at any point on the curve.
2. Evaluate the derivative at the point x = 1 to find the specific slope at (1, -1).
3. Use the point-slope form of a line to write the equation of the tangent line using the point (1, -1) and the slope obtained from step 2.

Firstly, let's find the derivative of the given function:

y' = d/dx (x² - 2x) = 2x - 2

Evaluating this derivative at x = 1 gives us:

y'(1) = 2(1) - 2 = 0

The slope of the tangent line at the point (1, -1) is 0. Therefore, the equation of the tangent line is simply:

y - (-1) = 0(x - 1)

Simplifying, we get the equation of the tangent line:

y = -1

This horizontal line touches the curve at the point (1, -1) and has a slope of 0.