Final answer:
To find the equation of the tangent line to the curve y = (x - x²) at x = 4, we need to find the slope of the tangent line at that point. By finding the derivative and evaluating it at x = 4, we get the slope of -7. Using the slope and the point (4, -12), we are able to determine the equation of the tangent line to be y = -7x + 16.
Step-by-step explanation:
To find the equation of the tangent line, we need to find the slope of the tangent line at the given point. The slope of a curve at a point can be found using the derivative of the function. So, first, we find the derivative of the function y = (x - x^2), which is dy/dx = 1 - 2x.
Substitute x = 4 into the derivative equation: dy/dx = 1 - 2(4) = 1 - 8 = -7.
Now we have the slope of the tangent line, which is -7. We also know that the equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. Using the point (4, y), we can substitute the values to find b: y = (-7)(4) + b, y = -28 + b.
Since the point (4, y) lies on the curve y = (x - x^2), we can substitute x = 4 into the equation and solve for y: y = 4 - 4^2, y = 4 - 16, y = -12.
Substituting the values of m = -7 and (x, y) = (4, -12) into the equation y = mx + b, we can solve for b: -12 = (-7)(4) + b, -12 = -28 + b, b = -12 + 28, b = 16.
Therefore, the equation of the tangent line to y = (x - x^2) at x = 4 is y = -7x + 16.