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1. (a) The corresponding parametric equations for the curve are x=tx=t, $y=\cos ...
Jan 15, 2025
Solution by Steps
step 1
The parametric equations for the curve are given as x=t x = t , y=cos⁡(πt) y = \cos(\pi t) , and z=sin⁡(πt) z = \sin(\pi t) . This indicates that the curve is a helix contained within a circular cylinder with its axis along the x x -axis
step 2
The first derivative of the position vector r(t)=ti+cos⁡(πt)j+sin⁡(πt)k \mathbf{r}(t) = t \mathbf{i} + \cos(\pi t) \mathbf{j} + \sin(\pi t) \mathbf{k} is calculated as follows: r′(t)=i−πsin⁡(πt)j+πcos⁡(πt)k \mathbf{r}^{\prime}(t) = \mathbf{i} - \pi \sin(\pi t) \mathbf{j} + \pi \cos(\pi t) \mathbf{k}
step 3
The second derivative of the position vector is given by: r′′(t)=−π2cos⁡(πt)j−π2sin⁡(πt)k \mathbf{r}^{\prime\prime}(t) = -\pi^2 \cos(\pi t) \mathbf{j} - \pi^2 \sin(\pi t) \mathbf{k}
Answer
The parametric equations describe a helix in a circular cylinder, with derivatives indicating the curve's behavior.
Key Concept
Parametric equations describe the position of points along a curve in space.
Explanation
The derivatives provide information about the curve's velocity and acceleration, which are essential for understanding its motion.
1. (a) The corresponding parametric equations for the curve are x=tx=t, y=cospit,z=sinpity=\\cos \\pi t, z=\\sin \\pi t. Since y2+z2=1y^{2}+z^{2}=1, the curve is contained in a circular cylinder with axis the xx-axis. Since x=tx=t, the curve is a helix.\n\\[\n\\text { (b) } \\begin{aligned}\n\\mathbf{r}(t) & =t \\mathbf{i}+\\cos \\pi t \\mathbf{j}+\\sin \\pi t \\mathbf{k} \\Rightarrow \\\\\n\\mathbf{r}^{\\prime}(t) & =\\mathbf{i}-\\pi \\sin \\pi t \\mathbf{j}+\\pi \\cos \\pi t \\mathbf{k} \\Rightarrow \\\\\n\\mathbf{r}^{\\prime \\prime}(t) & =-\\pi^{2} \\cos \\pi t \\mathbf{j}-\\pi^{2} \\sin \\pi t \\mathbf{k}\n\\end{aligned}\n\\]
Generated Graph
Solution by Steps
step 1
The parametric equations for the curve are given as x=t x = t , y=cos⁡(πt) y = \cos(\pi t) , and z=sin⁡(πt) z = \sin(\pi t) . This indicates that the curve is a helix contained within a circular cylinder with its axis along the x x -axis
step 2
The first derivative of the position vector is calculated as follows: r(t)=ti+cos⁡(πt)j+sin⁡(πt)k \mathbf{r}(t) = t \mathbf{i} + \cos(\pi t) \mathbf{j} + \sin(\pi t) \mathbf{k} Thus, r′(t)=i−πsin⁡(πt)j+πcos⁡(πt)k \mathbf{r}^{\prime}(t) = \mathbf{i} - \pi \sin(\pi t) \mathbf{j} + \pi \cos(\pi t) \mathbf{k}
step 3
The second derivative of the position vector is given by: r′′(t)=−π2cos⁡(πt)j−π2sin⁡(πt)k \mathbf{r}^{\prime \prime}(t) = -\pi^2 \cos(\pi t) \mathbf{j} - \pi^2 \sin(\pi t) \mathbf{k}
Answer
The curve is a helix described by the parametric equations, with derivatives indicating its behavior.
Key Concept
Parametric equations describe curves in space, and derivatives provide information about the curve's direction and curvature.
Explanation
The parametric equations define a helix, and the derivatives show how the position changes with respect to t t , indicating the curve's motion in three-dimensional space.
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