Final answer:
To find the equation of the tangent line to the curve at a given point, first compute the derivative to get the slope. Then, use the point-slope form with the given point and the slope to write the equation of the tangent line.
Step-by-step explanation:
The question is asking to find the equation of the tangent line to the curve y = 4x² - x³ at the point (1, 3). The first step is to find the derivative of the given function, which represents the slope of the tangent line. The derivative of y = 4x² - x³ is y' = 8x - 3x². Now we evaluate the derivative at the x-value of the given point, x = 1, to find the slope of the tangent line at that point, which is y'(1) = 8(1) - 3(1)² = 5. The slope of the tangent line is therefore 5. Since we have the slope and a point on the tangent line, we can use the point-slope form to write the equation of the tangent line: y - y1 = m(x - x1), plugging in (1, 3) for (x1, y1) and 5 for the slope m. This gives us the equation y - 3 = 5(x - 1), which simplifies to the final answer y = 5x - 2.