Final answer:
The gradient of the curve y = x^3 - 2x^2 + 5x - 3 at the point where it crosses the y-axis is found by differentiating the function and evaluating at x = 0, resulting in a gradient of 5.
Step-by-step explanation:
The student is asked to find the gradient of the curve y = x^3 - 2x^2 + 5x - 3 at the point where the curve crosses the y-axis. To find the gradient at a specific point, we first need to differentiate the function to find y' (the derivative of y with respect to x), which gives us the gradient of the tangent to the curve at any point x.
To find the gradient when the curve crosses the y-axis, we set x to 0 since all points on the y-axis satisfy x = 0. The given function is differentiated as follows: y' = 3x^2 - 4x + 5. Substituting x = 0 into y', we get y' = 5. Therefore, the gradient of the curve at the y-axis is 5.