Final answer:
To find the equation of the tangent line to the curve at the given point (2, 6), we differentiate the function y = x³ - 2x² with respect to x to find the slope of the tangent line. Plugging in the coordinates and the slope gives us the equation of the tangent line.
Step-by-step explanation:
To find the equation of the tangent line to the curve, we need to find the derivative of the curve at the given point (2, 6). The derivative represents the slope of the tangent line. To find the derivative of the curve y = x³ - 2x², we differentiate the function with respect to x. Differentiating y = x³ - 2x² gives us dy/dx = 3x² - 4x. Plugging in x = 2 gives us the slope of the tangent line as m = 3(2)² - 4(2) = 8. Now we can use the slope-intercept form of the equation of a line, y = mx + b, to find the equation of the tangent line. Plugging in the coordinates (2, 6) and the slope m = 8, we have 6 = 8(2) + b. Solving for b gives us b = -10. Therefore, the equation of the tangent line to the curve y = x³ - 2x² at the point (2, 6) is y = 8x - 10.