Let f(x,y) = y^2 e^{xy}. (a) Find the gradient of f. (b) Evaluate the gradient at point P(2,3). (c) Find the rate of change of f at P in the direction of the vector u = (-3/5, 4/5).
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First, we need to find the gradient of f. The gradient is a vector of partial derivatives with respect to each variable. In this case, we have two variables, I and y. So, we need to find the partial derivatives of f with respect to I and y. Show more…
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(a) Find the gradient of $ f $. (b) Evaluate the gradient at the point $ P $. (c) Find the rate of change of $ f $ at $ P $ in the direction of the vector $ u $. $ f(x, y) = x/y $, $ P(2, 1) $, $ u = \frac{3}{5} i + \frac{4}{5} j $
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(a) Find the gradient of $f$. (b) Evaluate the gradient at the point $P$. (c) Find the rate of change of $f$ at $P$ in the direction of the vector $\mathbf{u}$. $$ f(x, y)=x^{2} e^{y}, \quad P(3,0), \quad \mathbf{u}=\frac{1}{5}(3 \mathbf{i}-4 \mathbf{j}) $$
(a) Find the gradient of $f$. (b) Evaluate the gradient at the point $P$. (c) Find the rate of change of $f$ at $P$ in the direction of the vector $\mathbf{u}$. $$ f(x, y)=x / y, \quad P(2,1), \quad \mathbf{u}=\frac{3}{5} \mathbf{i}+\frac{4}{5} \mathbf{j} $$
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