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Chapter 11: Problem 20

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\left\langle\int_{0}^{t} \cos u^{2} d u, \int_{0}^{t} \sin u^{2} d u\right\rangle, t>0$$

Short Answer

Expert verified
Question: Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the given parameterized curve \(\mathbf{r}(t) = \int_{0}^{t} \cos u^{2} d u \mathbf{i} + \int_{0}^{t} \sin u^{2} d u \mathbf{j}\). Answer: The unit tangent vector, \(\mathbf{T}(t)\), is given by \(\mathbf{T}(t) = \frac{\left\langle \cos t^2, \sin t^2 \right\rangle}{\sqrt{(\cos t^2)^2 + (\sin t^2)^2}}\). The curvature, \(\kappa\), is calculated as \(\kappa = \frac{\|\mathbf{T}'(t)\|}{\|\mathbf{r}'(t)\|^3}\), where \(\|\mathbf{T}'(t)\|\) is the magnitude of \(\mathbf{T}'(t)\), which needs to be computed using differentiation.

Step by step solution

01

Find the derivative of the curve \(\mathbf{r}(t)\) with respect to \(t\)

Use the Fundamental Theorem of Calculus to differentiate the curve componentwise: $$ \mathbf{r}'(t) = \left\langle \frac{d}{dt} \int_{0}^{t} \cos u^{2} d u, \frac{d}{dt} \int_{0}^{t} \sin u^{2} d u \right\rangle = \left\langle \cos t^2, \sin t^2 \right\rangle $$
02

Find the magnitude of the tangent vector \(\|\mathbf{r}'(t)\|\)

The magnitude of the tangent vector is: $$ \|\mathbf{r}'(t)\| = \sqrt{(\cos t^2)^2 + (\sin t^2)^2} $$
03

Find the unit tangent vector \(\mathbf{T}(t)\)

The unit tangent vector is obtained by dividing the tangent vector \(\mathbf{r}'(t)\) by its magnitude: $$ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|} = \frac{\left\langle \cos t^2, \sin t^2 \right\rangle}{\sqrt{(\cos t^2)^2 + (\sin t^2)^2}} $$
04

Calculate the derivative of the unit tangent vector \(\mathbf{T}'(t)\)

Now we need to find the derivative of the unit tangent vector with respect to \(t\): $$ \mathbf{T}'(t) = \frac{d}{dt} \left\langle \frac{\cos t^2}{\sqrt{(\cos t^2)^2 + (\sin t^2)^2}}, \frac{\sin t^2}{\sqrt{(\cos t^2)^2 + (\sin t^2)^2}} \right\rangle $$ This can be a bit complicated, so it might be helpful to use a computer software to find the exact differentiation.
05

Find the magnitude of \(\mathbf{T}'(t)\) and the curvature \(\kappa\)

First, find the magnitude of \(\mathbf{T}'(t)\): $$ \|\mathbf{T}'(t)\| = \sqrt{(\frac{dT}{dt})^2} $$ Now, find the curvature \(\kappa\) using the following formula: $$ \kappa = \frac{\|\mathbf{T}'(t)\|}{\|\mathbf{r}'(t)\|^3} $$

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Most popular questions from this chapter

Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Geometric-arithmetic mean Use the vectors \(\mathbf{u}=\langle\sqrt{a}, \sqrt{b}\rangle\) and \(\mathbf{v}=\langle\sqrt{b}, \sqrt{a}\rangle\) to show that \(\sqrt{a b} \leq(a+b) / 2,\) where \(a \geq 0\) and \(b \geq 0\).

Cusps and noncusps a. Graph the curve \(\mathbf{r}(t)=\left\langle t^{3}, t^{3}\right\rangle .\) Show that \(\mathbf{r}^{\prime}(0)=\mathbf{0}\) and the curve does not have a cusp at \(t=0 .\) Explain. b. Graph the curve \(\mathbf{r}(t)=\left\langle t^{3}, t^{2}\right\rangle .\) Show that \(\mathbf{r}^{\prime}(0)=\mathbf{0}\) and the curve has a cusp at \(t=0 .\) Explain. c. The functions \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle\) and \(\mathbf{p}(t)=\left\langle t^{2}, t^{4}\right\rangle\) both satisfy \(y=x^{2} .\) Explain how the curves they parameterize are different. d. Consider the curve \(\mathbf{r}(t)=\left\langle t^{m}, t^{n}\right\rangle,\) where \(m>1\) and \(n>1\) are integers with no common factors. Is it true that the curve has a cusp at \(t=0\) if one (not both) of \(m\) and \(n\) is even? Explain.

Use the formula in Exercise 79 to find the (least) distance between the given point \(Q\) and line \(\mathbf{r}\). $$Q(-5,2,9) ; \mathbf{r}(t)=\langle 5 t+7,2-t, 12 t+4\rangle$$

Let \(\mathbf{u}(t)=\left\langle 1, t, t^{2}\right\rangle, \mathbf{v}(t)=\left\langle t^{2},-2 t, 1\right\rangle\) and \(g(t)=2 \sqrt{t}\). Compute the derivatives of the following functions. $$\mathbf{v}(g(t))$$

Let \(\mathbf{u}(t)=\left\langle 1, t, t^{2}\right\rangle, \mathbf{v}(t)=\left\langle t^{2},-2 t, 1\right\rangle\) and \(g(t)=2 \sqrt{t}\). Compute the derivatives of the following functions. $$\mathbf{v}\left(e^{t}\right)$$
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