Problem 30
Ask you to verify parts of Theorem \(11.2 .4 .\) In each let \(f(t)=t^{3}, \vec{r}(t)=\left\langle t^{2}, t-1,1\right\rangle\) and \(\vec{s}(t)=\) \(\left\langle\sin t, e^{t}, t\right\rangle .\) Compute the various derivatives as indicated. Simplify \(\vec{r}(t) \cdot \vec{s}(t),\) then find its derivative; show this is the same as \(\vec{r}^{\prime}(t) \cdot \vec{s}(t)+\vec{r}(t) \cdot \vec{s}^{\prime}(t)\)
Problem 31
Ask you to verify parts of Theorem \(11.2 .4 .\) In each let \(f(t)=t^{3}, \vec{r}(t)=\left\langle t^{2}, t-1,1\right\rangle\) and \(\vec{s}(t)=\) \(\left\langle\sin t, e^{t}, t\right\rangle .\) Compute the various derivatives as indicated. Simplify \(\vec{r}(t) \times \vec{s}(t),\) then find its derivative; show this is the same as \(\vec{r}^{\prime}(t) \times \vec{s}(t)+\vec{r}(t) \times \vec{s}^{\prime}(t)\)
Problem 31
Find the equation of the osculating circle to the curve at the indicated \(t\) -value. \(\vec{r}(t)=\left\langle t, t^{2}\right\rangle,\) at \(t=0\)
Problem 31
Find the average rate of change of \(\vec{r}(t)\) on the given interval. $$ \vec{r}(t)=\left\langle t, t^{2}\right\rangle \text { on }[-2,2] $$
Problem 31
Find the position function of an object given its acceleration and initial velocity and position. $$ \vec{a}(t)=\langle\cos t,-\sin t\rangle ; \quad \vec{v}(0)=\langle 0,1\rangle, \quad \vec{r}(0)=\langle 0,0\rangle $$
Problem 32
Ask you to verify parts of Theorem \(11.2 .4 .\) In each let \(f(t)=t^{3}, \vec{r}(t)=\left\langle t^{2}, t-1,1\right\rangle\) and \(\vec{s}(t)=\) \(\left\langle\sin t, e^{t}, t\right\rangle .\) Compute the various derivatives as indicated. Simplify \(\vec{r}(f(t))\), then find its derivative; show this is the same as \(\vec{r}^{\prime}(f(t)) f^{\prime}(t)\).
Problem 32
Find the average rate of change of \(\vec{r}(t)\) on the given interval. $$ \vec{r}(t)=\langle t, t+\sin t\rangle \text { on }[0,2 \pi] $$
Problem 32
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 \(t=0\)
Problem 32
Find the position function of an object given its acceleration and initial velocity and position. $$ \vec{a}(t)=\langle 0,-32\rangle ; \quad \vec{v}(0)=\langle 10,50\rangle, \quad \vec{r}(0)=\langle 0,0\rangle $$
Problem 33
Find the average rate of change of \(\vec{r}(t)\) on the given interval. $$ \vec{r}(t)=\langle 3 \cos t, 2 \sin t, t\rangle \text { on }[0,2 \pi] $$
