벡터와 공간 전체 문제

For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface.

${\rho }^{2}({\sin }^{2}(\varphi )-{\cos }^{2}(\varphi ))=1$

For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface.

${r}^{2}-2r\,\cos (\theta )+{z}^{2}=1$

If the boat velocity is $5$ km/h due north in still water and the water has a current of $2$ km/h due west (see the following figure), what is the velocity of the boat relative to shore? What is the angle $\theta$ that the boat is actually traveling?

When the boat reaches the shore, two ropes are thrown to people to help pull the boat ashore. One rope is at an angle of $25^\circ$ and the other is at $35^\circ .$ If the boat must be pulled straight and at a force of $500\text{N},$ find the magnitude of force for each rope (see the following figure).

An airplane is flying in the direction of 52° east of north with a speed of 450 mph. A strong wind has a bearing 33° east of north with a speed of 50 mph. What is the resultant ground speed and bearing of the airplane?

Calculate the work done by moving a particle from position $(1,2,0)$ to $(8,4,5)$ along a straight line with a force $\text{F}=2\text{i}+3\text{j}-\text{k}.$

Because your tire is flat, you are only able to apply your force at a $60^\circ$ angle. What is the torque at the center of the bolt? Assume this force is not enough to loosen the bolt.

Someone lends you a tire jack and you are now able to apply a 200-N force at an $80^\circ$ angle. Is your resulting torque going to be more or less? What is the new resulting torque at the center of the bolt? Assume this force is not enough to loosen the bolt.

Give the component functions $x=f(t)$ and $y=g(t)$ for the vector-valued function $\text{r}(t)=3\sec t\text{i}+2\tan t\text{j}.$

Given $\text{r}(t)=3\sec t\text{i}+2\tan t\text{j},$ find the following values (if possible).

a. $\text{r}\left(\frac{\pi }{4}\right)$ b. $\text{r}(\pi )$ c. $\text{r}\left(\frac{\pi }{2}\right)$

Sketch the curve of the vector-valued function $\text{r}(t)=3\sec t\text{i}+2\tan t\text{j}$ and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

Evaluate $\lim_{t\to 0}\langle {e}^{t},\frac{\sin t}{t},{e}^{-t}\rangle .$

Given the vector-valued function $\text{r}(t)=\langle \cos t,\sin t\rangle ,$ find the following values:

a. $\lim_{t\to \frac{\pi }{3}}\text{r}(t)$ b. $\text{r}\left(\frac{\pi }{3}\right)$ c. Is $\text{r}(t)$ continuous at $t=\frac{\pi }{3}?$ d. Graph $\text{r}(t).$

Given the vector-valued function $\text{r}(t)=\langle t,{t}^{2}+1\rangle ,$ find the following values:

a. $\lim_{t\to -3}\text{r}(t)$ b. $\text{r}(-3)$ c. Is $\text{r}(t)$ continuous at $t=-3?$ d. $\text{r}(t+2)-\text{r}(t)$

Let $\text{r}(t)={e}^{t}\text{i}+\sin t\text{j}+\ln t\text{k}.$ Find the following values:

a. $\text{r}\left(\frac{\pi }{4}\right)$ b. $\lim_{t\to \pi /4}\text{r}(t)$ c. Is $\text{r}(t)$ continuous at $t=\frac{\pi }{4}?$

Find the limit of the following vector-valued functions at the indicated value of t.

$\lim_{t\to 4}\langle \sqrt{t-3},\frac{\sqrt{t}-2}{t-4},\tan \left(\frac{\pi }{t}\right)\rangle$

Find the limit of the following vector-valued functions at the indicated value of t.

$\lim_{t\to \pi /2}\text{r}(t)$ for $\text{r}(t)={e}^{t}\text{i}+\sin t\text{j}+\ln t\text{k}$

Find the limit of the following vector-valued functions at the indicated value of t.

$\lim_{t\to \infty }\langle {e}^{-2t},\frac{2t+3}{3t-1},\arctan (2t)\rangle$

Find the limit of the following vector-valued functions at the indicated value of t.

$\lim_{t\to {e}^{2}}\langle t\ln (t),\frac{\ln t}{{t}^{2}},\sqrt{\ln ({t}^{2})}\rangle$

Find the limit of the following vector-valued functions at the indicated value of t.

$\lim_{t\to \pi /6}\langle {\cos }^{2}t,{\sin }^{2}t,1\rangle$