Chapter 11

Find the equation of the osculating circle to the curve at the indicated \(t\) -value. \(\vec{r}(t)=\langle 3 \cos t, \sin t\rangle,\) at \(t=\pi / 2\)

Evaluate the given definite or indefinite integral. $$ \int\left\langle t^{3}, \cos t, t e^{t}\right\rangle d t $$

Find the displacement, distance traveled, average velocity and average speed of the described object on the given interval. An object with position function \(\vec{r}(t)=\langle 2 \cos t, 2 \sin t, 3 t\rangle\), where distances are measured in feet and time is in seconds, on \([0,2 \pi]\).

Evaluate the given definite or indefinite integral. $$ \int\left\langle\frac{1}{1+t^{2}}, \sec ^{2} t\right\rangle d t $$

Find the average rate of change of \(\vec{r}(t)\) on the given interval. $$ \vec{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle \text { on }[-1,3] $$

Find the equation of the osculating circle to the curve at the indicated \(t\) -value. \(\vec{r}(t)=\left\langle t^{2}-t, t^{2}+t\right\rangle,\) at \(t=0\)

Find the displacement, distance traveled, average velocity and average speed of the described object on the given interval. An object with position function \(\vec{r}(t)=\langle 5 \cos t,-5 \sin t\rangle\), where distances are measured in feet and time is in seconds, on \([0, \pi]\).

Find the displacement, distance traveled, average velocity and average speed of the described object on the given interval. An object with velocity function \(\vec{v}(t)=\langle\cos t, \sin t\rangle,\) where distances are measured in feet and time is in seconds, on \([0,2 \pi]\)

Find the displacement, distance traveled, average velocity and average speed of the described object on the given interval. An object with velocity function \(\vec{v}(t)=\langle 1,2,-1\rangle,\) where distances are measured in feet and time is in seconds, on [0,10]

Solve the given initial value problems. Find \(\vec{r}(t),\) given that \(\vec{r}^{\prime}(t)=\langle t, \sin t\rangle\) and \(\vec{r}(0)=\langle 2,2\rangle\).