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True or False If a vector function \(\mathbf{r}= \mathbf{r}(t)\) is differentiable at \(t_{0}\), \(\ a<t_{0}<b\), and \(P_{0}\) is the point on the curve traced out by \(\mathbf{r}=\mathbf{r}(t)\) corresponding to \(t=t_{0},\) then the tangent line to the curve at \(t_{0}\) contains the point \(P_{0}\) and has the direction \(\mathbf{r}( t_{0}) \mathbf{.}\)

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True or False A curve traced out by a vector function \(\mathbf{r}=\mathbf{r}(t)\) is called a smooth curve on the interval \([ a,b]\) if \(\mathbf{r}=\mathbf{r}(t)\) is continuous on the interval \( [ a,b]. \)

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True or False If \(C\) is a smooth curve traced out by a vector function \(\mathbf{r}=\mathbf{r}(t)\), \(a\leq t\leq b\), then \(\mathbf{T}(t)= \mathbf{r}^{\prime} (t)\) is the unit tangent vector to \(C\) at \(t.\)

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True or False If \(C\) is a smooth curve traced out by a twice differentiable vector function \(\mathbf{r}=\mathbf{r}(t),\) \(a\leq t\leq b,\) and if \(\mathbf{T}\) is the unit tangent vector to \(C\), then \(\mathbf{N}( t) =\dfrac{\mathbf{T}( t) }{\left\Vert \mathbf{T}( t) \right\Vert }\) is the principal unit normal vector to \(C\) at \(t.\)

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True or False For a smooth space curve \(C\) traced out by \(\mathbf{r}=\mathbf{r}(t),\) \(a\leq t\leq b,\) the integral \(\int_{a}^{b}\left\Vert \mathbf{r}^{\prime} ( t) \right\Vert dt\) equals the arc length along \(C\) from \(t=a\) to \(t=b.\)

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True or False For a smooth curve traced out by a twice differentiable vector function, the unit tangent vector is orthogonal to the principal unit normal vector.

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\(\mathbf{r}(t)=t\mathbf{i}-t^{2}\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=t\mathbf{i}-t^{3}\mathbf{j,} \quad t=-1\)

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\(\mathbf{r}(t)=(t^{2}+1)\mathbf{i}+(1-t)\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=t^{3}\mathbf{i}+3t\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=4t\mathbf{i}-\sqrt{t}\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=\sqrt{t}\mathbf{i}+\dfrac{1}{2}t\mathbf{j,} \quad t=4\)

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\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j,} \quad t=0\)

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\(\mathbf{r}(t)=e^{2t}\mathbf{i}+e^{-t}\mathbf{j,} \quad t=0\)

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\(\mathbf{r}(t)=(1-3t)\mathbf{i}+2t\mathbf{j}-(5+t)\mathbf{k,} \quad t=0\)

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\(\mathbf{r}(t)=(2+t)\mathbf{i}+(2-t)\mathbf{j}+3t\mathbf{k,} \quad t=0\)

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\(\mathbf{r}(t)=\cos ( 2t) \mathbf{i}+\sin (2t) \mathbf{j}-5\mathbf{k,} \quad t=\dfrac{\pi }{4}\)

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\(\mathbf{r}(t)=3\mathbf{i}+\cos t\mathbf{j}+\sin t\mathbf{k,}\quad t=\dfrac{\pi }{6}\)

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\(\mathbf{r}(t)=2\cos t\mathbf{i}+\mathbf{j}+2\sin t\mathbf{k,}\quad t=\dfrac{\pi }{6}\)

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\(\mathbf{r}(t)=2\cos ( 2t) \mathbf{i}+2\sin ( 2t) \mathbf{j}+5\mathbf{k,} \quad t=\dfrac{\pi }{2}\)

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\(\mathbf{r}(t)=e^{t}\cos t\mathbf{i}+e^{t}\sin t\mathbf{j}+e^{t}\mathbf{k,} \quad t=0\)

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\(\mathbf{r}(t)=e^{-t}\cos t\mathbf{i}+e^{-t}\sin t\mathbf{j}-e^{-t}\mathbf{k,}\quad t=0 \)

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\(\mathbf{r}(t)=t\mathbf{i}+t^{2}\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=t\mathbf{i}-t^{3}\mathbf{j,} \quad t=-1\)

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\(\mathbf{r}(t)=(t^{2}+1)\mathbf{i}+(1-t)\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=t^{3}\mathbf{i}+3t\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=\dfrac{2t}{t+1}\mathbf{i}-\dfrac{t^{2}}{t+1}\mathbf{j,} \quad t=1\)

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\(\mathbf{r}(t)=\sqrt{t}\mathbf{i}+\dfrac{1}{2}t\mathbf{j}, \quad t=4\)

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\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j,} \quad t=0\)

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\(\mathbf{r}(t)=e^{2t}\mathbf{i}+e^{-t}\mathbf{j,} \quad t=0\)

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\(\mathbf{r}(t)=(1-3t)\mathbf{i}+2t\mathbf{j}-(5+t)\mathbf{k,} \quad t=0\)

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\(\mathbf{r}(t)=(2+t)\mathbf{i}+(2-t)\mathbf{j}+3t\mathbf{k,} \quad t=0\)

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\(\mathbf{r}(t)=2\cos t\mathbf{i}+\mathbf{j}+2\sin t\mathbf{k,}\quad t=\dfrac{\pi }{6}\)

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\(\mathbf{r}(t)=3\mathbf{i}+\cos t\mathbf{j}+\sin t\mathbf{k,} \quad t=\dfrac{\pi }{6}\)

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\(\mathbf{r}(t)=\cos ( 2t) \mathbf{i}+\sin (2t) \mathbf{j}-5\mathbf{k,} \quad t=\dfrac{\pi }{4}\)

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\(\mathbf{r}(t)=2\cos ( 2t) \mathbf{i}+2\sin ( 2t) \mathbf{j}+5\mathbf{k,} \quad t=\dfrac{\pi }{2}\)

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\(\mathbf{r}(t)=e^{t}\cos t\mathbf{i}+e^{t}\sin t\mathbf{j}+e^{t}\mathbf{k,}\quad t=0\)

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\(\mathbf{r}(t)=e^{-t}\cos t\mathbf{i}+e^{-t}\sin t\mathbf{j}-e^{-t}\mathbf{k,} \quad t=0\)

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\(\mathbf{r}(t)=t\mathbf{i}+(t^{3/2}+1)\mathbf{j}\) from \(t=1\) to \(t=8\)

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\(\mathbf{r}(t)=t\mathbf{i}+t^{3/2}\mathbf{j}\) from \(t=0\) to \(t=4\)

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\(\mathbf{r}(t)=8t\mathbf{i}+\dfrac{t^{6}+2}{t^{2}}\mathbf{j}\) from \(t=1\) to \(t=2\)

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\(\mathbf{r}(t)=t\mathbf{i}+(1-t^{2/3})^{3/2}\mathbf{j}\) from \(t= \dfrac{1}{8}\) to \(t=1\)

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\(\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+t\mathbf{k}\) from \(t=0\) to \(t=1\)

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\(\mathbf{r}(t)=2t\mathbf{i}+t\mathbf{j}+3t\mathbf{k}\) from \(t=1\) to \(t=3\)

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\(\mathbf{r}(t)=\sin ( 2t) \mathbf{i}+\cos (2t) \mathbf{j}+t\mathbf{k}\) from \(t=0\) to \(t=\pi \)

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\(\mathbf{r}(t)=\sin t\mathbf{i}+\cos t\mathbf{j}+bt\mathbf{k}\) from \(t=0\) to \(t=2\pi \)

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\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\sqrt{2}t\mathbf{k}\) from \(t=0\) to \(t=1\)

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\(\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}+\mathbf{k}\) from \(t=0\) to \(t=\dfrac{\pi }{2}\)

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\(\mathbf{r}(t)=e^{t}\cos t\mathbf{i}+e^{t}\sin t\mathbf{j}+e^{t}\mathbf{k}\) from \(t=\) \(0\) to \(t=2\pi \)

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\(\mathbf{r}(t)=t^{2}\mathbf{i}-2\sqrt{2}\ t\mathbf{j}+(t^{2}-1)\mathbf{k}\) from \(t=0\) to \(t=1\)

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\(\mathbf{r}( t) =3\sin t\mathbf{i}-4\cos t\mathbf{j}\) from \(t=0\) to \(t=\pi\)

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\(\mathbf{r}( t) =5\cos t\mathbf{i}+3\sin t\mathbf{j}\) from \(t=0\) to \(t=\dfrac{\pi }{2}\)

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\(\mathbf{r}( t) =2\sin t\mathbf{i}+4\cos (2t) \mathbf{j}\) from \(t=0\) to \(t= {2\pi}\)

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\(\mathbf{r}( t) =4\sin ( 2t) \mathbf{i}-6\cos t\,\mathbf{j}\) from \(t=0\) to \(t=\dfrac{\pi }{2}\)

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Find all the points on the curve traced out by \(\mathbf{r} (t)=t^{2}\mathbf{i}+(t^{2}-1)\mathbf{j}-t\mathbf{k}\) at which \(\mathbf{r}(t)\) and its tangent vector are orthogonal.

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Find the angle between the helix traced out by \(\mathbf{r}( t) =\cos t\mathbf{i}+t\mathbf{j}+\sin t\mathbf{k}\) and the \(y\)-axis.

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The helix defined by \(\mathbf{r}(t)=3\sin t\mathbf{i}+3\cos t\mathbf{j}+4t\mathbf{k}\) and the direction \(\mathbf{k}\) meet at a constant angle. Find the angle to the nearest degree.

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The helix defined by \(\mathbf{r}(t)=2\cos t\mathbf{i}+2\sin t\mathbf{j}+t\mathbf{k}\) and the direction \(\mathbf{k}\) meet at a constant angle. Find the angle to the nearest degree.

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\(\mathbf{r}_{1}(t_{1})=t_{1}\mathbf{i}+\sin \left( \pi t_{1}\right) \mathbf{j}+\mathbf{k}\) and \(\mathbf{r}_{2}(t_{2})=\mathbf{i}+t_{2}\mathbf{j}+\left( 1+t_{2}\right) \mathbf{k}\) at \(\left( 1,0,1\right)\)

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\(\mathbf{r}_{1}(t_{1})=\sin \left( 2t_{1}\right) \mathbf{i}+\sin t_{1}\mathbf{j}+t_{1}\mathbf{k}\) and \(\mathbf{r}_{2}(t_{2})=t_{2}^{2}\mathbf{i}+t_{2}\mathbf{j}+t_{2}^{3}\mathbf{k}\) at \(\left( 0,0,0\right) \)

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\(\mathbf{r}_{1}(t_{1})=(e^{t_{1}}-1)\mathbf{i}-\cos \left( \pi t_{1}\right) \mathbf{j}+t_{1}\mathbf{k}\) and \(\mathbf{r}_{2}(t_{2})=(1-t_{2})\mathbf{i}-\mathbf{j}+(t_{2}-1)\mathbf{k}\) at \((0,-1,~0)\)

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\(\mathbf{r}_{1}(t_{1})=t_{1}^{2}\mathbf{i}-( t_{1}-3) \mathbf{j}-( t_{1}-3) \mathbf{k}\) and \(\mathbf{r}_{2}(t_{2})=\big(t_{2}^{2}+3\big)\mathbf{i}-(t_{2}-2) \mathbf{j}+t_{2}\mathbf{k}\) at \((4,1,1)\)

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\(\mathbf{r}(t)=t^{2}\mathbf{i}-( t-3) \mathbf{j}-( t-3) \mathbf{k}\)

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\(\mathbf{r}(t)=\left( t+5\right) \mathbf{i}-\left( t+5\right) \mathbf{j}+t^{2}\mathbf{k}\)

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\(\mathbf{r}(t)=\cos t\mathbf{i}+\sin t\mathbf{j}+t\mathbf{k}\)

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\(\mathbf{r}(t)=2\sin t\mathbf{i}+2\cos t\mathbf{j}+3t\mathbf{k}\)

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Suppose the vector function \(\mathbf{r=r}(t)\) in space is differentiable and has a nonzero tangent vector at \(t_{0}\). Find the direction cosines for the tangent vector at \(t_{0}\).

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Show that at any point, the tangent vectors to a helix \(\mathbf{r}(t)=a\cos t\mathbf{i}+a\sin t\mathbf{j}+bt\mathbf{k}\), where \(a\) and \(b\) are real numbers, intersect the direction of the \(z\)-axis at a constant angle.

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Use Simpson's Rule with \(n=4\) to approximate the length of the curve \(\mathbf{r}(t)=t^{2}\mathbf{i}+t^{3}\mathbf{j}+(2t+3)\mathbf{k}\), \(0\leq t\leq 2\).

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1. Approximate the arc length of \(\mathbf{r}( t) =3\sin t\mathbf{i}-4\cos t\mathbf{j}\) from \(t=0\) to \(t=\pi \) using Simpson's Rule with \(n=4.\)
2. Approximate the arc length of \(\mathbf{r}( t)=3\sin t\mathbf{i}-4\cos t\mathbf{j}\) from \(t=0\) to \(t=\pi \) using the Trapezoidal Rule with \(n=6.\)
3. Compare the answers to (a) and (b) to the result given by technology (see Problem 55).

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1. Approximate the arc length of \(\mathbf{r}( t) =2\sin t\mathbf{i}+4\cos ( 2t)\mathbf{j}\) from \(t=0\) to \(t=2\pi\) using Simpson's Rule with \(n=4.\)
2. Approximate the arc length of \(\mathbf{r}( t) =2\sin t\mathbf{i}+4\cos ( 2t)\mathbf{j}\) from \(t=0\) to \(t=2\pi\) using the Trapezoidal Rule with \(n=6.\)
3. Compare the answers to (a) and (b) to the result given by technology (see Problem 57).

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Approximate the length of the curve \(\mathbf{r}(t)=\dfrac{1}{3}t^{3}\mathbf{i}+(t-1)\mathbf{j}+2\mathbf{k}\), \(0\leq t\leq \dfrac{1}{2}\), correct to three decimal places, by expanding the integrand in a power series.