Final answer:
To find the equation of the tangent line to the curve y = 6x cos x at the point (π, -6π), we calculate the derivative, evaluate it at x = π to get the slope -6, and then use the point-slope form to find the tangent line equation: y = -6x + 12π.
Step-by-step explanation:
To find an equation of the tangent line to the curve y = 6x cos x at the given point (π, -6π), we need to calculate the derivative of the function to find the slope of the tangent line at that point. The derivative of y with respect to x, using the product and chain rules, is:
y' = d/dx (6x cos x) = 6 cos x - 6x sin x
At x = π, we substitute π into the derivative to get the slope of the tangent line:
y'(π) = 6 cos(π) - 6π sin(π) = -6
Now we use the point-slope form of a line to write the equation of the tangent line:
y - y1 = m(x - x1), where (x1, y1) is the point on the curve and m is the slope.
For the given point (π, -6π) and slope -6, the equation of the tangent line is:
y - (-6π) = -6(x - π)
Simplifying this, we get:
y = -6x + 12π