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
Step 1: Find the y-coordinate of the point on the curve. Given the curve y = 4x^2 - 6x + 5 and x=1. Substitute x=1 into the equation to find the corresponding y-value: y = 4(1)^2 - 6(1) + 5 y = 4 - 6 + 5 y = 3 So, the point on the curve is (1, 3). Step 2: Find the gradient of the tangent to the curve. Differentiate the curve's equation with respect to x to find the gradient function (dy)/(dx): (dy)/(dx) = (d)/(dx)(4x^2 - 6x + 5) (dy)/(dx) = 8x - 6 Now, evaluate the gradient of the tangent at x=1: m_tangent = 8(1) - 6 m_tangent = 8 - 6 m_tangent = 2 Step 3: Find the gradient of the normal to the curve. The normal is perpendicular to the tangent. If m_tangent is the gradient of the tangent, then the gradient of the normal m_normal is given by m_normal = -(1)/(m_tangent). m_normal = -(1)/(2) Step 4: Find the equation of the normal. Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), with the point (x_1, y_1) = (1, 3) and the gradient m = -(1)/(2): y - 3 = -(1)/(2)(x - 1) Multiply both sides by 2 to eliminate the fraction: 2(y - 3) = -1(x - 1) 2y - 6 = -x + 1 Rearrange the equation into the standard form Ax + By + C = 0: x + 2y - 6 - 1 = 0 x + 2y - 7 = 0 The equation of the normal to the curve at x=1 is x + 2y - 7 = 0. Last free one today — make it count tomorrow, or type /upgrade for unlimited.