Final answer:
To find the equation of the tangent line at x=2 for the function f(x) = 8x³ - 2x² + 2x, we need to find the slope of the tangent line at that point and use the point-slope form of a linear equation.
Step-by-step explanation:
To find the equation of the tangent line at x=2 for the function f(x) = 8x³ - 2x² + 2x, we need to find the slope of the tangent line at that point and use the point-slope form of a linear equation.
To find the slope, we need to find the derivative of the function and evaluate it at x=2. Taking the derivative, we get f'(x) = 24x² - 4x + 2. Evaluating at x=2, we get f'(2) = 24(2)² - 4(2) + 2 = 88.
So the slope of the tangent line at x=2 is 88. Now we can use the point-slope form, which is y - y₁ = m(x - x₁), where (x₁, y₁) is the point (2, f(2)). Substituting the values, we get y - f(2) = 88(x - 2). Simplifying, we get the equation of the tangent line as y = 88x - 172.