[tex]f(x,y)=\dfrac xy[/tex]
a. The gradient is
[tex]\nabla f(x,y)=\dfrac1y\,\vec\imath-\dfrac x{y^2}\,\vec\jmath[/tex]
b. At the point (9, 1), the gradient has value
[tex]\nabla f(9,1)=\vec\imath-9\,\vec\jmath[/tex]
c. The derivative of [tex]f[/tex] at (9, 1) in the direction of [tex]\vec u=\dfrac35\,\vec\imath+\dfrac45\,\vec\jmath[/tex] is
[tex]D_uf(9,1)=\nabla f(9,1)\cdot\dfrac{\vec u}{\|\vec u\|}[/tex]
We have
[tex]\|\vec u\|=\sqrt{\left(\dfrac35\right)^2+\left(\dfrac45\right)^2}=1[/tex]
so
[tex]D_uf(9,1)=(\vec\imath-9\,\vec\jmath)\cdot\left(\dfrac35\,\vec\imath+\dfrac45\,\vec\jmath\right)=-\dfrac{33}5[/tex]
The rate of change of f at P in the direction of the vector u is 7.8
Given the function [tex]f(x, y) =\frac{x}{y}[/tex] at the point P(9, 1) where the vector [tex]\vec{u}=\frac{3}{5}i+\frac{4}{5}j[/tex]
a) To get the gradient of "f"
[tex]\nabla f(x, y)=\frac{\delta f}{\delta x}i + \frac{\delta f}{\delta y}j\\ \nabla f(x, y)=\frac{\delta (x/y)}{\delta x}i + \frac{\delta (x/y)}{\delta y}j\\\nabla f(x, y)= \frac{1}{y}i+(\frac{-x}{y^2} )j\\\nabla f(x, y)= \frac{1}{y}i-\frac{x}{y^2} j\\[/tex]
b) Substituting x = 9 and y = 1 into the result in (a), we will have:
[tex]\nabla f(x, y)= \frac{1}{y}i-\frac{x}{y^2} j\\\nabla f(9, 1)= \frac{1}{1}i-\frac{9}{1^2} j\\\nabla f(9, 1)= i-9j[/tex]
c) We need to get the derivative of ∇f(9, 1) in the direction of the vector [tex]\vec{u}=\frac{3}{5}i+\frac{4}{5}j[/tex]
The formula for the derivative in the direction is expressed as;
[tex]D_uf(9,1)=\nabla f(9, 1) \cdot \vec{u}=1-9j \cdot \frac{\vec{u}}{\mid {u}\mid}[/tex]
Get the modulus of the vector |u|
[tex]\mid{u}\mid=\sqrt{(\frac{3}{5})^2+(\frac{4}{5})^2 } \\\mid{u}\mid=\sqrt{\frac{9}{25} +\frac{16}{25} }\\\mid{u}\mid=\sqrt{\frac{25}{25} }\\\mid{u}\mid=1[/tex]
Substitute the given parameters into the derivative formula:
[tex]D_uf(9,1)=1-9j \cdot \frac{\frac{3}{5}i-\frac{4}{5}j }{1}\\D_uf(9,1)=1(\frac{3}{5}) + (-9)(\frac{-4}{5} ) )\\D_uf(9,1)=\frac{3}{5}+\frac{36}{5} \\D_uf(9,1)=\frac{39}{5}\\D_uf(9,1)= 7.8[/tex]
Hence the rate of change of f at P in the direction of the vector u is 7.8
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