Final answer:
To calculate the magnitude of the gradient of the function f(x,y) at P=(9,81), one should find the partial derivatives and then evaluate them at the point P, before using these evaluations to find the magnitude of the gradient vector.
Step-by-step explanation:
The question asks to calculate the magnitude of the gradient (denoted as ‘∇fP’) of the function f(x,y)=xe^{x²}-y at the point P=(9,81).
First, we find the partial derivatives of the function to get the components of the gradient vector:
∇f(x,y) = (∂f/∂x, ∂f/∂y)
∂f/∂x = e^{x²} + 2x^2e^{x²}
∂f/∂y = -1
Next, we evaluate these partial derivatives at the point P=(9,81):
∂f/∂x at P = e^{81} + 2*9^2e^{81}
= e^{81} + 2*81e^{81}
= 163e^{81}
∂f/∂y at P = -1
Now, we use these components to calculate the magnitude of the gradient at P:
∇fP = (163e^{81}, -1)
‖∇fP‖ = √((163e^{81})^2 + (-1)^2)
Finally, we find the magnitude, which will provide us with the answer.
Since the calculation involves very large numbers because of the term e^{81}, in a real scenario, only a scientific calculator or computer software capable of handling such large numbers would be able to give an accurate value.