This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the equation of the normal to the curve y = 4x^3 - 5x^2 + x at the point (1, 2), we follow these steps: Step 1: Find the derivative of the curve to get the slope of the tangent. Given the curve: y = 4x^3 - 5x^2 + x Differentiate y with respect to x: (dy)/(dx) = (d)/(dx)(4x^3) - (d)/(dx)(5x^2) + (d)/(dx)(x) (dy)/(dx) = 12x^2 - 10x + 1 Step 2: Calculate the slope of the tangent at the given point (1, 2). Substitute x=1 into the derivative: m_tangent = 12(1)^2 - 10(1) + 1 m_tangent = 12 - 10 + 1 m_tangent = 3 (You can use an fx-991 calculator to find the derivative at x=1 by inputting (d)/(dx)(4x^3 - 5x^2 + x)|_x=1, which will give 3.) Step 3: Determine the slope of the normal. The normal line is perpendicular to the tangent line. The product of their slopes is -1. m_normal = -(1)/(m_tangent) m_normal = -(1)/(3) Step 4: Use the point-slope form to find the equation of the normal. The point is (x_1, y_1) = (1, 2) and the slope is m_normal = -(1)/(3). The point-slope form is y - y_1 = m(x - x_1). y - 2 = -(1)/(3)(x - 1) Step 5: Rearrange the equation to match the given options. Multiply both sides by 3 to eliminate the fraction: 3(y - 2) = -1(x - 1) 3y - 6 = -x + 1 Add 6 to both sides: 3y = -x + 1 + 6 3y = -x + 7 This matches Option A. The final answer is 3y = -x + 7. That's 2 down. 3 left today — send the next one.