# Thread: compute the directional derivatives

1. ## compute the directional derivatives

Hi!
There is an exam on Tuesday.
I have no idea of these problems.

Compute the directional derivatives of the following functions in the directions indicated.

(a) x²+xy-xz+2y at (2,1,0) in the direction in which the directional derivative is a maximum.

(b) e^xsiny (as a function of two variables) at (0,∏/6) in the direction making a +60˚ angle with the positive x-axis.

(c) x²+2y²-3x+y at (-2,4) in the direction of the tangent (from left to right)of the curve discribed by y=x²

(d) e^xyz+x²+y² at (1,1,1) in the direction of the curve described by x=t, y=2t²-1, z=t³

(e) x²-y²+z² in the direction of the outer normal of ghe surface x²+y²+z=7 at (2,-1,2)

2. If we have a scalar field (acts like a function) $\displaystyle \phi(x_n,y_n,z_n)=\phi(P_n)\,,\,P_n=(x_n,y_n,z_n)$, we define its gradient as

$\displaystyle \nabla \phi(P_n)=\frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y} \hat{j} + \frac{\partial \phi}{\partial z}\hat{k}$.

The gradient will always point in the direction where the function increases the most.

The directional derivative in the direction of $\displaystyle \vec{u}$ then becomes

$\displaystyle D_u \phi(P_n)=\nabla \phi (P_n)\cdot \vec{u}$

where $\displaystyle \vec{u}$ is a unit vector.

Now you just have to compute $\displaystyle \frac{\vec{v}}{|\vec{v}|}=\vec{u}$ for all the problems below.

3. For (b), a unit vector in the direction making angle $\displaystyle \theta$ with the positive x-axis is $\displaystyle cos(\theta)\vec{i}+ sin(\theta)\vec{j}$.

4. For this solution, what is the vector u in problems (d) and (e) ??

5. (d) e^xyz+x²+y² at (1,1,1) in the direction of the curve described by x=t, y=2t²-1, z=t³
What is the tangent vector to this curve at (1, 1, 1)?

(e) x²-y²+z² in the direction of the outer normal of the surface x²+y²+z=7 at (2,-1,2)
What is the normal vector to this surface at (2, -1, 2)?