14. Directional Derivatives and Gradients

Homework

1. Find the derivative of the function, \(f(x,y,z)=x\sin z+y\cos z\), along the curve, \(r(t)=(3t^2,2t^3,\pi t)\), at \(t=1\). Use two methods.

   1. Use the gradient formula.

   2. Differentiate the composition.

2. Find the derivative of the function, \(f(x,y,z)=ze^x+ye^z\), along a curve, \(\vec r(t)\), at the point, \(\vec r(1)=(1,2,3)\), with velocity, \(\vec v(1)=\langle4,2,1\rangle\).

3. A weather balloon is located at \(P=(x,y,z)=(400,200,100)\,\text{mi}\) and has velocity \(\vec v=\langle 20,30,20\rangle\,\dfrac{\text{mi}}{\text{hr}}\). It measures the temperature to be \(T|_P=27^\circ\,\text{C}\) and it partial derivatives to be \[ \dfrac{\partial T}{\partial x}=0.4\,\dfrac{^\circ\text{C}}{\text{mi}} \qquad \dfrac{\partial T}{\partial y}=0.2\,\dfrac{^\circ\text{C}}{\text{mi}} \qquad \dfrac{\partial T}{\partial z}=0.3\,\dfrac{^\circ\text{C}}{\text{mi}} \] Find the rate of change of the temperature as seen by the balloon at \(P\).

4. Find the derivative of the function, \(f(x,y,z)=xz+y+xyz\), along the vector, \(\vec v=\langle2,-3,4\rangle\), at the point, \(P=(-2,3,5)\).

5. Find the derivative of the function, \(f(x,y,z)=xz+y+xyz\), at the point, \(P=(-2,3,5)\), in the direction toward the point, \(Q=(-1,5,3)\).

6. The altitude of a mountain is given by \[ a=100-\dfrac{x^2}{9}-\dfrac{y^2}{16} \] Find the slope of the land at \((x,y)=(3,4)\) in the direction of the vector \(\vec v=\langle 4,3\rangle\).

7. Duke Skywater is aboard the Centurian Eagle at the point \(P=(2,3,-1)\) in the middle of a deadly polaron field whose density is \(D=yz-xz+xy\).

   1. Find the derivative of the polaron density in the direction of the vector \(\vec u=\langle4,-12,3\rangle\).

   2. If Duke is currently traveling with velocity \(\vec v=\langle-3,4,12\rangle\), what is the rate of change of the polaron density as seen by Duke?

   3. If Duke wants to change his direction to escape the polaron field as soon as possible, in what unit vector direction, \(\hat w\), should he travel to reduce the polaron density as fast as possible?

   4. If the maximum speed of the Centurian Eagle is, \(|\vec w|=6\sqrt{2}\), what is the maximum rate at which he can reduce the polaron density?
      (The answer should be negative.)

8. Find all points \(P=(p,q,r)\) on the hyperboloid \(36x^2+9y^2-4z^2=36\) where the normal vector is parallel to the vector \(\vec v=\langle3,3,-2\rangle\).

9. In \(\mathbb R^3\), consider the point \(P=(1,2,3)\) on the level surface \(F(x,y,z)=xyz+3x+2y+z=16\).

   1. Find the normal vector at \(P\).

   2. Find the normal form of the tangent plane at \(P\).

   3. Find the parametric equation of the normal line at \(P\).

10. In \(\mathbb R^2\), consider the point \(P=(1,2)\) on the level curve \(F(x,y)=xy+2x+y=6\).

    1. Normal vector at point \(P\).

    2. Find the normal line at \(P\), in parametric and standard forms.

    3. Find the tangent line at \(P\), in parametric and standard forms.
       (Note: If \(\vec n=\langle n_1,n_2\rangle\) is a normal vector, then \(\vec v=\vec n^\perp=\langle n_2,-n_1\rangle\) is a tangent vector.)