Directional Derivatives And Gradient Vector

(requires JavaScript)

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