Answer: To find the gradient of a curve at a given point, we need to find the derivative of the function with respect to x and then evaluate it at the desired x-coordinate.
Given the function y = 3x^4 - 2x^2 + 5x - 2, we can find its derivative dy/dx as follows:
dy/dx = d/dx(3x^4) - d/dx(2x^2) + d/dx(5x) - d/dx(2)
Taking the derivative of each term:
dy/dx = 12x^3 - 4x + 5
Now, let's evaluate the derivative at the given points:
Point (0, -2):
Substituting x = 0 into the derivative:
dy/dx = 12(0)^3 - 4(0) + 5
dy/dx = 0 - 0 + 5
dy/dx = 5
Therefore, the gradient at (0, -2) is 5.
Point (1, 4):
Substituting x = 1 into the derivative:
dy/dx = 12(1)^3 - 4(1) + 5
dy/dx = 12 - 4 + 5
dy/dx = 13
Therefore, the gradient at (1, 4) is 13.