In Problems 1–4, for each vector function \(\mathbf{r}=\mathbf{r} ( t) \)
\(\mathbf{r}(t)=t^{2}\mathbf{i}+3t\mathbf{j,}\;t=2\)
\(\mathbf{r}(t)=3\cos t\mathbf{i}+\sin t\mathbf{j,}\;0\leq t\leq \pi, \;t=\dfrac{\pi }{3}\)
\(\mathbf{r}(t)=t\mathbf{i}+2\cos t\mathbf{j}+2\sin t\mathbf{k,}\;t=0\)
\(\mathbf{r}(t)=t\mathbf{i}+t\mathbf{j}-2\mathbf{k},\;t=1\)
Find \(\lim\limits_{t\rightarrow 3}\left[ ( t-3) \mathbf{i}+ \dfrac{t^{2}-3t}{t^{2}-9}\mathbf{j}-\dfrac{4}{t}\mathbf{k}\right]\).
In Problems 6 and 7, determine if the vector function is continuous at each number in the given interval. Identify any numbers where \(\mathbf{r} =\mathbf{r}(t)\) is discontinuous.
\(\mathbf{r}( t)\;=\;\dfrac{t}{t-2}\mathbf{i}+\sin t \mathbf{j},\) interval \(( 0,2\pi ) \)
\(\mathbf{r}( t)\;=\;\dfrac{t}{t+1}\mathbf{i}+\sqrt{t-1}\mathbf{j}+3t \mathbf{k}\), interval \([ 1,3] \)
In Problems 8 and 9, find \({\bf r}^{\prime} (t)\) and \( {\bf r}^{\prime \prime} (t)\).
\(\mathbf{r}(t)=3t\mathbf{i}-\ln t\mathbf{j}+2e^{t}\mathbf{k}\)
\(\mathbf{r}(t)=2\cos t\mathbf{i}+3\cos t\mathbf{j}+t\mathbf{k}\)
In Problems 10 and 11, find \([{\bf f}(t)\,{\cdot}\, {\bf g} (t)]^{\prime} \) and \([{\bf f}(t)\times {\bf g}(t)]^{\prime} \).
\(\mathbf{f}(t)=t\mathbf{i}-\dfrac{t}{2}\mathbf{j}+\mathbf{k}\) and \(\mathbf{g}(t)=\sqrt{1-t}\mathbf{i}+t^{3}\mathbf{j}+2( t+1) \mathbf{k}\)
\(\mathbf{f}(t)=t\mathbf{i}+\cos ( 2t) \mathbf{j} -5 \mathbf{k}\) and \(\mathbf{g}(t)=2t\mathbf{i}+\cos t\mathbf{j}+\sin t\mathbf{k}\)
In Problems 12–14, for each curve \(C\) traced out by \(\mathbf{r}=\mathbf{r} ( t) \):
\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\mathbf{k}\)
\(\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+\sqrt{1-5t^{2}} \mathbf{k}\), \(\dfrac{-\sqrt{5}}{5}< t< \dfrac{\sqrt{5}}{5}\)
\(\mathbf{r}(t)=e^{t}\sin t\mathbf{i}+e^{t}\cos t\mathbf{j}+e^{t}\mathbf{k}\)
Find the acute angle between the tangent vector to the helix traced out by \(\mathbf{r} ( t)\;=\;\sin t\mathbf{i}+3t\mathbf{j}+\cos t\mathbf{k}\) and the direction \(\mathbf{j}\).
In Problems 16–18, find the arc length of each vector function.
\(\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}\) from \(t=\;0\) to \(t=2\pi \)
\(\mathbf{r}(t)=t\mathbf{i} +2t\mathbf{j}+\sqrt{1-5t^{2}}\mathbf{k}\) from \(t=0\) to \(t=\dfrac{{1}}{4 }\)
\(\mathbf{r}(t)=\cos t\mathbf{i}+\sin t\mathbf{j}+\dfrac{t}{5} \mathbf{k}\) from \(t=\;0\) to \(t=2\pi \)
In Problems 19 and 20, determine whether the parameter is arc length.
\(\mathbf{r}(t)=3t\mathbf{i}+4t\mathbf{j}\; +\left( 5t+1\right) \mathbf{k}\)
\(\mathbf{r}(t)=3\cos t\mathbf{i}+3\sin t\mathbf{j}\;+8t\mathbf{k}\)
In Problems 21–23, find the curvature \(\kappa\;=\;\kappa (t)\) of the curve \(C\) traced out by each vector function \({\bf r}={\bf r}( t) \).
\(\mathbf{r}(t)=4\cos t\mathbf{i}-4\sin t\mathbf{j}\)
\(\mathbf{r}(t)=t\mathbf{i}+\cos t \mathbf{j}\;+\sin t\mathbf{k}\)
\(\mathbf{r}(t)=e^{t}\mathbf{i}+4t\mathbf{j}\;+5e^{-t}\mathbf{k}\)
Find the curvature of the graph of \(y=x^{3}-4\) at the point \(( 1,-3)\).
Find the curvature of the graph of \(y=4e^{-2x}\) at the point \(( 0,4)\).
In Problems 26–28, find the radius of the osculating circle at the point corresponding to \(t\) on the curve \(C\) traced out by each vector function.
\(\mathbf{r}(t)=( 1-t^{2}) \mathbf{i}+e^{-t}\mathbf{j, }\;t=0\)
\(\mathbf{r}(t)=\cos ^{2}t \mathbf{i}+\sin ^{2}t\mathbf{j},\;t=\dfrac{\pi }{3}\)
\(\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+\sqrt{1-t^{2}}\mathbf{k} ,\;-1< t< \;1,\;t=0\)
Find the curvature of the curve \(y=\sqrt[3]{x}, x>0\).
Find the minimum curvature of the curve \(C\) traced out by the vector function \(\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}\).
In Problems 31–34:
\(\mathbf{r}(t)=2\cos t\mathbf{i}+\sin t\mathbf{j}\)
\(\mathbf{r}(t)=e^{t}\sin t\mathbf{i} +e^{-t}\mathbf{j}\)
\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\mathbf{k}\)
\(\mathbf{r}(t)=e^{t}\sin t \mathbf{i}+e^{t}\cos t\mathbf{j}+e^{t}\mathbf{k}\)
Find the velocity and acceleration of a particle moving on the cycloid \(\mathbf{r}( t)\;=\;[ \pi t-\sin ( \pi t) ] \mathbf{i} +[ 1-\cos ( \pi t) ] \mathbf{j}\).
In Problems 37 and 38, find each integral.
\(\int [ ( t^{2}-2) \mathbf{i}-( t-2) ^{2}\mathbf{j}+e^{2t}\mathbf{k}] dt\)
\(\int \left(\cos \dfrac{t}{2}\mathbf{i}+\sin ^{2}\left( \dfrac{t}{2}\right) \mathbf{j}+3t\mathbf{k}\right)dt\)
In Problems 39–41, solve each vector differential equation with the given boundary condition.
\(\mathbf{r}^{\prime} ( t)\;=\;e^{2t}\mathbf{i}+\ln t \mathbf{j}+2e^{t}\mathbf{k,}\;\mathbf{r}(1)=\mathbf{j}+\mathbf{k}\)
\(\mathbf{r}^{\prime} (t)=\sin t\cos t\mathbf{i}+\tan t\sec t \mathbf{j}+t\mathbf{k,}\;\mathbf{r}(0)=\mathbf{i}+\mathbf{k}\)
\(\dfrac{d\mathbf{r}}{dt}=\cos t\mathbf{i}-\sin t\mathbf{j} +\dfrac{t}{2}\mathbf{k,}\;\mathbf{r}( 0)\;=\;\mathbf{i}\)
Find the speed, velocity, and position at time \(t\) for a particle whose acceleration at time \(t\) is given by \(\mathbf{a}(t)=\tan t\sec t\mathbf{i}+\cos {t}\mathbf{j}\). Assume that the initial velocity and position are \(\mathbf{0}\).
Projectile Motion A projectile is fired with a speed of \(500{\textrm m}/\!{\textrm s}\) at an inclination of \(45° %TCIMACRO{\U{b0}}% %BeginExpansion {{}^\circ}% %EndExpansion \) to the horizontal from a point \(30{\textrm m}\) above level ground. Find the point where the projectile strikes the ground.
State Kepler's three laws of Planetary Motion. Explain each law in your own words.