Essay 10 points
Calculate the curvature of the function .

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Essay 10 points Calculate the curvature of the function $$ \vec{r}(t)=<\cos \left(t^{2}\right), \sin \left(t^{2}\right), 0> $$. Show all steps. Edit View Insert Format Tools Table p (t) Previous

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1. Compute the first derivative  
   The given vector-valued function is  
   $$
   \vec{r}(t)=\langle \cos(t^2),\, \sin(t^2),\, 0 \rangle.
   $$  
   Differentiating component‐wise with respect to $$t$$:
   - For the $$x$$-component:  
     $$
     \frac{d}{dt} \cos(t^2) = -\sin(t^2)\cdot 2t = -2t\,\sin(t^2).
     $$
   - For the $$y$$-component:  
     $$
     \frac{d}{dt} \sin(t^2) = \cos(t^2)\cdot 2t = 2t\,\cos(t^2).
     $$
   - For the $$z$$-component:  
     $$
     \frac{d}{dt} 0 = 0.
     $$
   Thus,  
   $$
   \vec{r}'(t)=\langle -2t\,\sin(t^2),\; 2t\,\cos(t^2),\; 0 \rangle.
   $$

2. Compute the second derivative  
   Differentiate $$\vec{r}'(t)$$ component by component.
   - For the $$x$$-component:  
     $$
     \frac{d}{dt} \bigl[-2t\,\sin(t^2)\bigr] = -2\,\sin(t^2) - 2t\cdot\cos(t^2)\cdot 2t = -2\sin(t^2) - 4t^2\cos(t^2).
     $$
   - For the $$y$$-component:  
     $$
     \frac{d}{dt} \bigl[2t\,\cos(t^2)\bigr] = 2\,\cos(t^2) - 2t\cdot\sin(t^2)\cdot 2t = 2\cos(t^2) - 4t^2\sin(t^2).
     $$
   - For the $$z$$-component:  
     $$
     \frac{d}{dt}(0) = 0.
     $$
   Thus,  
   $$
   \vec{r}''(t)=\langle -2\sin(t^2)-4t^2\cos(t^2),\; 2\cos(t^2)-4t^2\sin(t^2),\; 0 \rangle.
   $$

3. Compute the cross product $$\vec{r}'(t) \times \vec{r}''(t)$$  
   Write  
   $$
   \vec{r}'(t)=\langle -2t\,\sin(t^2),\; 2t\,\cos(t^2),\; 0 \rangle,
   $$
   $$
   \vec{r}''(t)=\langle -2\sin(t^2)-4t^2\cos(t^2),\; 2\cos(t^2)-4t^2\sin(t^2),\; 0 \rangle.
   $$
   The cross product is computed using the determinant formula:
   $$
   \vec{r}'(t) \times \vec{r}''(t) =
   \begin{vmatrix}
   \mathbf{i} & \mathbf{j} & \mathbf{k} \
   -2t\,\sin(t^2) & 2t\,\cos(t^2) & 0 \
   -2\sin(t^2)-4t^2\cos(t^2) & 2\cos(t^2)-4t^2\sin(t^2) & 0
   \end{vmatrix}.
   $$
   Since the first two components lie in the $$xy$$-plane, the cross product will only have a $$ \mathbf{k} $$-component:
   $$
   [\vec{r}'(t) \times \vec{r}''(t)]_k = (-2t\,\sin(t^2))\bigl(2\cos(t^2)-4t^2\sin(t^2)\bigr) - (2t\,\cos(t^2))\bigl[-2\sin(t^2)-4t^2\cos(t^2)\bigr].
   $$
   Expanding each term:
   - First term:
     $$
     -2t\,\sin(t^2)\cdot 2\cos(t^2) = -4t\,\sin(t^2)\cos(t^2),
     $$
     $$
     -2t\,\sin(t^2)\cdot (-4t^2\sin(t^2)) = +8t^3\,\sin^2(t^2).
     $$
   - Second term:
     $$
     -\, 2t\,\cos(t^2)\cdot (-2\sin(t^2)) = +4t\,\cos(t^2)\sin(t^2),
     $$
     $$
     -\, 2t\,\cos(t^2)\cdot (-4t^2\cos(t^2)) = +8t^3\,\cos^2(t^2).
     $$
   Combining the terms:
   $$
   -4t\,\sin(t^2)\cos(t^2) + 8t^3\,\sin^2(t^2) + 4t\,\sin(t^2)\cos(t^2) + 8t^3\,\cos^2(t^2).
   $$
   The terms $$-4t\,\sin(t^2)\cos(t^2)$$ and $$+4t\,\sin(t^2)\cos(t^2)$$ cancel, yielding:
   $$
   8t^3\left[\sin^2(t^2) + \cos^2(t^2)\right] = 8t^3.
   $$
   Therefore,
   $$
   \vec{r}'(t) \times \vec{r}''(t) = \langle 0,\; 0,\; 8t^3 \rangle.
   $$

4. Compute the magnitudes  
   The magnitude of the cross product is:
   $$
   ||\vec{r}'(t) \times \vec{r}''(t)|| = |8t^3| = 8|t^3|.
   $$
   Next, compute $$||\vec{r}'(t)||$$:
   $$
   ||\vec{r}'(t)|| = \sqrt{(-2t\,\sin(t^2))^2 + (2t\,\cos(t^2))^2 + 0^2} = \sqrt{4t^2\sin^2(t^2) + 4t^2\cos^2(t^2)}.
   $$
   Factor out $$4t^2$$:
   $$
   ||\vec{r}'(t)|| = \sqrt{4t^2\left[\sin^2(t^2) + \cos^2(t^2)\right]} = \sqrt{4t^2} = 2|t|.
   $$

5. Compute the curvature  
   The formula for curvature is:
   $$
   \kappa(t)=\frac{||\vec{r}'(t) \times \vec{r}''(t)||}{||\vec{r}'(t)||^3}.
   $$
   Substituting the values:
   $$
   \kappa(t)=\frac{8|t^3|}{(2|t|)^3}=\frac{8|t^3|}{8|t|^3} = 1.
   $$

The curvature of the function is therefore  
$$
\kappa(t)=1.
$$
The curvature of the function $$ \vec{r}(t)=\langle \cos(t^2), \sin(t^2), 0 \rangle $$ is $$ \kappa(t) = 1 $$.

Essay 10 points
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Essay 10 points Calculate the curvature of the function . Show all steps. Edit View Insert Format Tools Table p (t) Previous

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Essay 10 points
Calculate the curvature of the function .

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Edit View Insert Format Tools Table
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