Find the position vector-valued function $\mathbf { r } ( t )$, given that $\mathbf { a } ( t ) = \mathbf { i } + e ^ { t } \mathbf { j }$, $\mathbf { v } ( 0 ) = 2 \mathbf { j }$, and $\mathbf { r } ( 0 ) = 2 \mathbf { i }$.

Determine the values (if any) of $t$ for which the function is continuous.

$$\mathbf{r}(t)=\sqrt{t} \mathbf{i}+\frac{1}{t-4} \mathbf{j}+\mathbf{k}$$

Find a vector function r that satisfies the indicated conditions. $\mathbf{r}^{\prime \prime}(t)=12 t \mathbf{i}-3 t^{-1 / 2} \mathbf{j}+2 \mathbf{k} ; \mathbf{r}^{\prime}(1)=\mathbf{j}, \mathbf{r}(1)=2 \mathbf{i}-\mathbf{k}$

Solve the initial value problems to the exercise given below for r as a vector function of t.

Differential equation: $\quad \frac{d \mathbf{r}}{d t}=\left(t^3+4 t\right) \mathbf{i}+t \mathbf{j}+2 t^2 \mathbf{k}$

Initial condition: $\mathbf{r}(0)=\mathbf{i}+\mathbf{j}$

Find the domain of the vector function.

$$\mathbf{r}(t)=\cos t\mathbf{i}+2\sin t\mathbf{j}+\sqrt{t+1}\mathbf{k}$$

Given $T: P_1 \rightarrow R^2$ be the function defined by the formula

$$T(p(x))=(p(0), p(1))$$

Determine $T^{-1}(3,5)$, and sketch its graph.

Find the position function from the given velocity or acceleration function. $\mathbf{v}(t)=\left\langle 12 \sqrt{t}, t /\left(t^{2}+1\right), t e^{-t}\right\rangle, \mathbf{r}(0)=\langle 8,-2,1\rangle$

In this exercise, determine the interval(s) on which the vector-valued function is continuous.

$$\mathbf{r}(t)=\left\langle e^t \sqrt{t}, \tan t\right\rangle$$

Determine where each vector function is continuous. $\mathbf{r}(t)=\sin t \mathbf{i}-\tan t \mathbf{j}+\mathbf{k}$

Find the derivative of the given vector-valued function. $\mathbf{r}(t)=\left\langle t^{4}, \sqrt{t+1}, \frac{3}{t^{2}}\right\rangle$

Find the domain of the vector-valued function.

$$\mathbf{r}(t)=\left\langle\frac{1}{t+1}, \frac{t}{2}\right\rangle$$

Find the derivative of the given vectorvalued function. $\mathbf{r}(t)=\left\langle\sqrt{t^{2}+1}, \sin 4 t, \ln 4 t\right\rangle$

Find the domain of each vector function. $\mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin t \mathbf{j}+2 t \mathbf{k}$

Find the velocity, acceleration, and speed of a particle with the given position function.

$$\mathbf{r}(t)=\cosh t \mathbf{i}+\sinh t \mathbf{j}+t \mathbf{k}$$