Final answer:
To calculate the magnitude of ∥∇fP∥, we need to find the gradient of f(x, y) at point P(2, 4) and then find its magnitude using the formula ∥∇fP∥ = √(∂f/∂x)^2 + (∂f/∂y)^2.
Step-by-step explanation:
To calculate the magnitude of ∞f(P), we need to find the gradient of f(x, y) at point P(2, 4) and then find its magnitude.
The gradient of f(x, y) is given by ∈f = (∂f/∂x, ∂f/∂y). So, we need to find ∂f/∂x and ∂f/∂y, and substitute the values x = 2 and y = 4 into those expressions.
Once we have ∂f/∂x and ∂f/∂y, we can find ∞f(P) using the formula ∞f(P) = (∂f/∂x, ∂f/∂y) and then find its magnitude using the formula ||∞f(P)|| = √(∂f/∂x)^2 + (∂f/∂y)^2.
To calculate ‖∇fP‖, we must first compute the gradient of the function f(x,y)=xex²−y at point P=(2,4). The gradient of f is a vector of partial derivatives, so we calculate ∂f/∂x and ∂f/∂y. The partial derivative with respect to x is ex² + 2x²ex² and with respect to y is −1. Evaluating these at P yields ∇fP = (e4 + 2*2²e4, -1) or ∇fP = (e4(1+16), -1). We can then find the magnitude ‖∇fP‖ by calculating the square root of the sum of the squares of these components: ‖∇fP‖ = √[(e4(1+16))2 + (-1)2]. The magnitude of the gradient at P is thus ‖∇fP‖ = √[e8*172 + 1].