Question

Consider the following. f(x, y) = x/y, P(9, 1), u = 3 5 i + 4 5 j (a) Find the gradient of f. ∇f(x, y) = (b) Evaluate the gradient at the point P. ∇f(9, 1) = (c) Find the rate of change of f at P in the direction of the vector u. Duf(9, 1) =

Answer 1

$f(x,y)=\dfrac xy$

a. The gradient is

$\nabla f(x,y)=\dfrac1y\,\vec\imath-\dfrac x{y^2}\,\vec\jmath$

b. At the point (9, 1), the gradient has value

$\nabla f(9,1)=\vec\imath-9\,\vec\jmath$

c. The derivative of $f$ at (9, 1) in the direction of $\vec u=\dfrac35\,\vec\imath+\dfrac45\,\vec\jmath$ is

$D_uf(9,1)=\nabla f(9,1)\cdot\dfrac{\vec u}{\|\vec u\|}$

We have

$\|\vec u\|=\sqrt{\left(\dfrac35\right)^2+\left(\dfrac45\right)^2}=1$

so

$D_uf(9,1)=(\vec\imath-9\,\vec\jmath)\cdot\left(\dfrac35\,\vec\imath+\dfrac45\,\vec\jmath\right)=-\dfrac{33}5$