벡터함수의 미분과 적분

Let $\text{r}(t)=t\text{i}+{t}^{2}\text{j}-{t}^{4}\text{k}$ and $\text{s}(t)=\sin (t)\text{i}+{e}^{t}\text{j}+\cos (t)\text{k}.$ Here is the graph of the function:

Find the following.

$\frac{d}{dt}[\text{r}(t)\cdot \text{s}(t)]$

Compute the first, second, and third derivatives of $\text{r}(t)=3t\text{i}+6\ln (t)\text{j}+5{e}^{-3t}\text{k}.$

Let $\text{r}(t)=t\text{i}+{t}^{2}\text{j}-{t}^{4}\text{k}$ and $\text{s}(t)=\sin (t)\text{i}+{e}^{t}\text{j}+\cos (t)\text{k}.$ Here is the graph of the function:

Find the following.

Find $\text{r}'(t)\cdot \text{r}''(t)\,\text{for}\,\text{r}(t)=-3{t}^{5}\text{i}+5t\text{j}+2{t}^{2}\text{k}.$

The acceleration function, initial velocity, and initial position of a particle are

$\text{a}(t)=-5\cos t\text{i}-5\sin t\text{j},\text{v}(0)=9\text{i}+2\text{j},\,\text{and}\,\text{r}(0)=5\text{i}.$

Find $\text{v}(t)\,\text{and}\,\text{r}(t).$

The position vector of a particle is $\text{r}(t)=5\sec (2t)\text{i}-4\tan (t)\text{j}+7{t}^{2}\text{k}.$

a. Graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function. b. Find the velocity as t approaches but is not equal to $\pi /4$ (if it exists).

Find the velocity and the speed of a particle with the position function $\text{r}(t)=\left(\frac{2t-1}{2t+1}\right)\text{i}+\ln (1-4{t}^{2})\text{j}.$ The speed of a particle is the magnitude of the velocity and is represented by $\| {r}^{'}(t)\| .$

Find the velocity function and show that $\text{v}(t)$ is always orthogonal to $\text{r}(t).$

Show that the speed of the particle is proportional to the angular velocity.

A particle moves on a circular path of radius b according to the function $\text{r}(t)=b\cos (\omega t)\text{i}+b\sin (\omega t)\text{j},$ where $\omega$ is the angular velocity, $d\theta /dt.$

Evaluate $\frac{d}{dt}[\text{u}(t)\,\times \,{u}^{'}(t)]$ given $\text{u}(t)={t}^{2}\text{i}-2t\text{j}+\text{k}.$

Find the antiderivative of $\text{r}'(t)=\cos (2t)\text{i}-2\sin t\text{j}+\frac{1}{1+{t}^{2}}\text{k}$ that satisfies the initial condition $\text{r}(0)=3\text{i}-2\text{j}+\text{k}.$

A particle moves on a circular path of radius b according to the function $\text{r}(t)=b\cos (\omega t)\text{i}+b\sin (\omega t)\text{j},$ where $\omega$ is the angular velocity, $d\theta /dt.$

Evaluate ${\int }_{0}^{3}\| t\text{i}+{t}^{2}\text{j}\| dt.$

An object starts from rest at point $P(1,2,0)$ and moves with an acceleration of $\text{a}(t)=\text{j}+2\text{k},$ where $\| \text{a}(t)\|$ is measured in feet per second per second. Find the location of the object after $t=2$ sec.

Show that if the speed of a particle traveling along a curve represented by a vector-valued function is constant, then the velocity function is always perpendicular to the acceleration function.

A particle moves on a circular path of radius b according to the function $\text{r}(t)=b\cos (\omega t)\text{i}+b\sin (\omega t)\text{j},$ where $\omega$ is the angular velocity, $d\theta /dt.$

Given $\text{r}(t)=t\text{i}+3t\text{j}+{t}^{2}\text{k}$ and $\text{u}(t)=4t\text{i}+{t}^{2}\text{j}+{t}^{3}\text{k},$ find $\frac{d}{dt}(\text{r}(t)\,\times \,\text{u}(t)).$

Given $\text{r}(t)=\langle t+\cos t,t-\sin t\rangle ,$ find the velocity and the speed at any time.

A particle moves on a circular path of radius b according to the function $\text{r}(t)=b\cos (\omega t)\text{i}+b\sin (\omega t)\text{j},$ where $\omega$ is the angular velocity, $d\theta /dt.$

Find the velocity vector for the function $\text{r}(t)=\langle {e}^{t},{e}^{-t},0\rangle .$

Find an equation of the tangent line to the curve $\text{r}(t)=\langle {e}^{t},{e}^{-t},0\rangle$ at $t=0.$

Describe and sketch the curve represented by the vector-valued function $\text{r}(t)=\langle 6t,6t-{t}^{2}\rangle .$

Locate the highest point on the curve $\text{r}(t)=\langle 6t,6t-{t}^{2}\rangle$ and give the value of the function at this point.

The position vector for a particle is $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+{t}^{3}\text{k}.$ The graph is shown here:

Find the velocity vector at any time.