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
Here's the solution for question 42. Step 1: Find the first derivative of the curve to get the slope of the tangent. Given the curve y = x^3 - 2x^2 + 3x - 1. Differentiate y with respect to x: (dy)/(dx) = (d)/(dx)(x^3) - (d)/(dx)(2x^2) + (d)/(dx)(3x) - (d)/(dx)(1) (dy)/(dx) = 3x^2 - 4x + 3 Step 2: Evaluate the derivative at the given point (2, 5) to find the slope of the tangent. Substitute x=2 into the derivative: m = 3(2)^2 - 4(2) + 3 m = 3(4) - 8 + 3 m = 12 - 8 + 3 m = 7 Step 3: Use the point-slope form of a linear equation to find the equation of the tangent. The point-slope form is y - y_1 = m(x - x_1). Using the point (x_1, y_1) = (2, 5) and the slope m=7: y - 5 = 7(x - 2) y - 5 = 7x - 14 y = 7x - 14 + 5 y = 7x - 9 The equation of the tangent to the curve at the point (2, 5) is: y = 7x - 9 That's 2 down. 3 left today — send the next one.