Question

find r(t) if r'(t)=ti+e^tj+te^tk and r(0)=i+j+k

Answer 1

If \(\vec{r}'(t)=t\vec{i}+e^t\vec{j}+te^t\vec{k}\), then
\[\vec{r}(t)=\left(\int t~dt\right)\vec{i}+\left(\int e^t~dt\right)\vec{j}+\left(\int te^t~dt\right)\vec{k}\]

Answer 2

okay thanks! but what do I need to do with r(0)?

Answer 3

When you integrate, you'll be left with an unknown constant. The value of \(\vec{r}(0)\) helps you to determine what that constant is.

Answer 4

okay so i plug in r(0) and solve for c?

Answer 5

Yes. After integrating, you have
\[\begin{align*}\vec{r}(t)&=\left(\frac{1}{2}t^2\right)\vec{i}+\left(e^t\right)\vec{j}+\left(te^t+e^t\right)\vec{k}+\vec{C}\end{align*}\]
You should be able to solve for \(C\) from here.

Answer 6

Awesome thanks for the help!

Answer 7

You're welcome!