### Directional Derivatives And Gradient Vector

(requires JavaScript)

1. Find the directional derivative of $f\left(x,y\right)=\sqrt{5x-4y}$ at the point $\left(4,1\right)$ in the direction corresponding to the angle $\theta =-\frac{\pi }{6}$.
$\frac{5}{16}\sqrt{3}+\frac{1}{4}$
1. Find the gradient of $f\left(x,y\right)=5x{y}^{2}-4{x}^{3}y$,
2. Evaluate the gradient at the point $P\left(1,2\right)$,
3. Find the rate of change of $f$ at $P$ in the direction of the vector $⟨\frac{5}{13},\frac{12}{13}⟩$.
1. $\nabla f\left(x,y\right)=⟨5{y}^{2}-12{x}^{2}y,10xy-4{x}^{3}⟩$
2. $⟨-4,16⟩$
3. $\frac{172}{13}$
1. Find the gradient of $f\left(x,y,z\right)=\sqrt{x+yz}$,
2. Evaluate the gradient at the point $P\left(1,3,1\right)$,
3. Find the rate of change of $f$ at $P$ in the direction of the vector $⟨\frac{2}{7},\frac{3}{7},\frac{6}{7}⟩$.
1. $\frac{⟨1,z,y⟩}{2\sqrt{x+yz}}$
2. $\frac{⟨1,1,3⟩}{4}$
3. $\frac{23}{28}$
2. Find the directional derivative of the function $f\left(x,y,z\right)=\frac{x}{y+z}$ at the point $\left(4,1,1\right)$ in the direction of the vector $⟨1,2,3⟩$.
$\frac{-9}{2\sqrt{14}}$.
3. Find the maximum rate of change of $f\left(x,y\right)=\frac{{y}^{2}}{x}$ at the point $\left(2,4\right)$ and the direction in which it occurs.
$4\sqrt{2}$ and $⟨-1,1⟩$.
4. Find the maximum rate of change of $f\left(x,y,z\right)=\mathrm{tan}\left(x+2y+3z\right)$ at the point $\left(-5,1,1\right)$ and the direction in which it occurs.
$\sqrt{14}$, $⟨1,2,3⟩$.
5. Find the directions in which the directional derivative of $f\left(x,y\right)={x}^{2}+\mathrm{sin}\left(xy\right)$ at the point $\left(1,0\right)$ has the value $1$.
$⟨0,1⟩$ and $⟨\frac{4}{5},-\frac{3}{5}⟩$.
6. Find the equations of the tangent plane and the normal line to the surface ${x}^{2}-2{y}^{2}+{z}^{2}+yz=2$ at the point $\left(2,1,-1\right)$.
$4x-5y-z=4$ and $\frac{x-2}{4}=\frac{y-1}{-5}=\frac{z+1}{-1}$
7. Find the equations of the tangent plane and the normal line to the surface $x-z=4{\mathrm{tan}}^{-1}\left(yz\right)$ at the point $\left(1+\pi ,1,1\right)$.
8. Let $f\left(x,y\right)={x}^{2}+4{y}^{2}$. Find the gradient vector $\nabla f\left(2,1\right)$ and use it to find the tangent line to the level curve $f\left(x,y\right)=8$ at the point $\left(2,1\right)$. Sketch the level curve, the tangent line, and the gradient vector.
$⟨4,8⟩$ and $x+2y=4$.
9. Find the points on the ellipsoid ${x}^{2}+2{y}^{2}+3{z}^{2}=1$ where the tangent plane is parallel to the plane $3x-y+3z=1$.
$±⟨\frac{-3\sqrt{2}}{5},\frac{\sqrt{2}}{10},\frac{-\sqrt{2}}{5}⟩$.
10. Find parametric equations for the tangent line to the curve of intersection of the paraboloid $z={x}^{2}+{y}^{2}$ and the ellipsoid $4{x}^{2}+{y}^{2}+{z}^{2}=9$ at the point $\left(-1,1,2\right)$.
$x=-1-10t,\phantom{\rule{0.5em}{0ex}}y=1-16t,\phantom{\rule{0.5em}{0ex}}z=2-12t$