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: Expand the given function y = (x-2)(2x^2-3x-5). y = x(2x^2-3x-5) - 2(2x^2-3x-5) y = 2x^3 - 3x^2 - 5x - 4x^2 + 6x + 10 y = 2x^3 - 7x^2 + x + 10 Step 2: Differentiate y with respect to x to find (dy)/(dx). Using the power rule for differentiation, (d)/(dx)(ax^n) = nax^n-1: (dy)/(dx) = (d)/(dx)(2x^3) - (d)/(dx)(7x^2) + (d)/(dx)(x) + (d)/(dx)(10) (dy)/(dx) = 2(3)x^3-1 - 7(2)x^2-1 + 1x^1-1 + 0 (dy)/(dx) = 6x^2 - 14x + 1 Step 3: Find the point where the curve cuts the y-axis. The curve cuts the y-axis when x=0. Substitute x=0 into the original equation for y: y = (0-2)(2(0)^2-3(0)-5) y = (-2)(-5) y = 10 So, the curve cuts the y-axis at the point (0, 10). Step 4: Find the gradient of the tangent at the point where the curve cuts the y-axis. Substitute x=0 into the derivative (dy)/(dx) to find the gradient: (dy)/(dx)|_x=0 = 6(0)^2 - 14(0) + 1 (dy)/(dx)|_x=0 = 0 - 0 + 1 (dy)/(dx)|_x=0 = 1 The gradient of the tangent to the curve where it cuts the y-axis is 1. 2 left today. What's next?