Question

Find the standard slopeintercept form ) equation of the tangent line to the following: f(x)=x^(2)-3x+1 at x=1

Answer 1

To find the tangent line to the curve f(x)=x²-3x+1 at x=1, we calculate the derivative, find the slope at that point, use the point-slope form, and convert to slope-intercept form, resulting in the equation y = -x + 1.

Step-by-step explanation:

To find the standard slope-intercept form equation of the tangent line to the function f(x)=x²-3x+1 at x=1, we need to follow these steps:

1. Find the derivative of f(x), which gives us the slope of the tangent line at any point on the curve.
2. Evaluate the derivative at x=1 to find the slope of the tangent line at that point.
3. Use the point-slope form of the equation of a line to find the equation of the tangent line.
4. Convert the equation into the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

Step 1, the derivative of f(x) is f'(x) = 2x - 3.
Step 2, the slope at x=1 is f'(1) = 2(1) - 3 = -1.
Step 3, the point on the curve at x=1 has coordinates (1, f(1)), which is (1, -1).
Step 4, using the point-slope form, the equation is y - (-1) = -1(x - 1), and simplifying gives us the slope-intercept form: y = -x + 1.