WeBWorK Standalone Renderer

The gradient vector $\nabla f(a,b)$ at a point $P = (a,b)$ and several unit vectors $\vec{r}, \vec{s}, \vec{t}, \vec{u}, \vec{n},$ and $\vec{e}$ in the $xy$ -plane are shown in the figure. You may assume that angles that look the same are the same. Here, $f_{\vec{v}} \, (a,b)$ denotes the directional derivative, which is often written as $D_{\vec{v}} \, f(a,b)$ .

Are the following statements true or false?

1. $f_{\vec{r}} \, (a,b) = f_{\vec{t}} \, (a,b).$

2. $f_{\vec{e}} \, (a,b) = \langle 0, 0 \rangle.$

3. $f_{\vec{r}} \, (a,b) < f_{\vec{s}} \, (a,b).$

4. $f_{\vec{r}} \, (a,b) < f_{\vec{u}} \, (a,b).$

5. $f_{\vec{n}} \, (a,b) = || \nabla f(a,b) ||.$

6. $f_{\vec{s}} \, (a,b)$ points in the direction of minimum rate of change of $f.$

7. $f_{\vec{r}} \, (a,b) < 0.$

8. $f_{\vec{u}} \, (a,b) = f_{\vec{t}} \, (a,b).$

(Click on graph to enlarge)

You can earn 50% partial credit for 4 - 5 correct answers, and 75% partial credit for 6 - 7 correct answers.