Question

Find the equation of the tangent line to the curve f(x) = x² - 3x + 6 at (1, 4).

Answer 1

To find the equation of the tangent line at (1, 4) for the curve f(x) = x² - 3x + 6, calculate the derivative to get the slope (-1) at x = 1 and use the point-slope form, resulting in the equation y = -x + 5.

Step-by-step explanation:

To find the equation of the tangent line to the curve f(x) = x² - 3x + 6 at the point (1, 4), we need to determine the slope of the tangent line and use the point-slope form of a line.

1. First, calculate the derivative of f(x) to find the slope of the tangent line at any point x. The derivative f'(x) = 2x - 3.
2. Then, calculate the slope at the point x = 1 by plugging it into the derivative: f'(1) = 2(1) - 3 = -1. This is the slope of the tangent line at (1, 4).
3. To write the equation of the line, we use the point-slope formula: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Plugging in the slope and the point (1, 4), we get y - 4 = -1(x - 1).
4. Simplify this equation to get the standard form of the equation of the tangent line: y = -x + 5.