/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Find the unit tangent vector \(\... [FREE SOLUTION] | 91影视

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Chapter 12: 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
Solution: The unit tangent vector \(\mathbf{T}\) for the given parameterized curve is: $$\mathbf{T} = \left\langle \cos t^{2},\sin t^{2}\right\rangle$$ The curvature \(\kappa\) for the given parameterized curve is: $$\kappa = 2t$$

Step by step solution

01

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

To find \(\mathbf{r'}(t)\), we need to differentiate the components of \(\mathbf{r}(t)\) with respect to \(t\). By the fundamental theorem of calculus, we have: $$\mathbf{r'}(t)=\left\langle\frac{d}{dt} \int_{0}^{t} \cos u^{2} d u, \frac{d}{dt} \int_{0}^{t} \sin u^{2} d u\right\rangle$$ Using the fundamental theorem of calculus and the chain rule, we can now differentiate the integrals: $$\mathbf{r'}(t) = \left\langle \cos t^{2},\sin t^{2}\right\rangle$$
02

Compute the magnitude of \(\mathbf{r'}(t)\)

To find the unit tangent vector \(\mathbf{T}\), we need to divide \(\mathbf{r'}(t)\) by its magnitude. So, first, let's find the magnitude of \(\mathbf{r'}(t)\): $$|\mathbf{r'}(t)| = \sqrt{(\cos t^{2})^{2}+(\sin t^{2})^{2}} = \sqrt{\cos^2t^2 + \sin^2t^2}$$ Here, we can use the identity \(\cos^2 x + \sin^2 x = 1\) to simplify the expression to: $$|\mathbf{r'}(t)| = \sqrt{1} = 1$$
03

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

Now that we have found the magnitude of \(\mathbf{r'}(t)\), we can find the unit tangent vector \(\mathbf{T}\) by dividing \(\mathbf{r'}(t)\) by its magnitude: $$\mathbf{T}=\frac{\mathbf{r'}(t)}{|\mathbf{r'}(t)|} = \frac{\left\langle \cos t^{2},\sin t^{2}\right\rangle}{1} = \left\langle \cos t^{2},\sin t^{2}\right\rangle$$
04

Compute the derivative of \(\mathbf{T}(t)\) with respect to \(t\)

To find the curvature \(\kappa\), we need to compute the magnitude of the derivative of the unit tangent vector with respect to \(t\). So, let's differentiate \(\mathbf{T}(t)\): $$\mathbf{T'}(t) = \left\langle -2t\sin t^{2},2t\cos t^{2}\right\rangle$$
05

Compute the magnitude of \(\mathbf{T'}(t)\)

Now, let's find the magnitude of \(\mathbf{T'}(t)\): $$|\mathbf{T'}(t)| = \sqrt{(-2t\sin t^{2})^{2}+(2t\cos t^{2})^{2}}$$ Further simplifying, we get: $$|\mathbf{T'}(t)| = 2t\sqrt{(\sin t^{2})^{2}+(\cos t^{2})^{2}} = 2t\sqrt{\sin^2t^2 + \cos^2t^2} = 2t \cdot 1 = 2t$$
06

Compute the curvature \(\kappa\)

Finally, we have all the necessary information to compute the curvature \(\kappa\). We can use the expression: $$\kappa = \frac{|\mathbf{T'}(t)|}{|\mathbf{r'}(t)|}$$ Plugging in the values for \(|\mathbf{T'}(t)|\) and \(|\mathbf{r'}(t)|\): $$\kappa = \frac{2t}{1} = 2t$$ So, the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the given parameterized curve are: $$\mathbf{T} = \left\langle \cos t^{2},\sin t^{2}\right\rangle$$ $$\kappa = 2t$$

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Most popular questions from this chapter

A race Two people travel from \(P(4,0)\) to \(Q(-4,0)\) along the paths given by $$ \begin{aligned} \mathbf{r}(t) &=(4 \cos (\pi t / 8), 4 \sin (\pi t / 8)\rangle \text { and } \\\ \mathbf{R}(t) &=\left(4-t,(4-t)^{2}-16\right) \end{aligned} $$ a. Graph both paths between \(P\) and \(Q\) b. Graph the speeds of both people between \(P\) and \(Q\) c. Who arrives at \(Q\) first?

Evaluate the following limits. $$\lim _{t \rightarrow \ln 2}\left(2 e^{t} \mathbf{i}+6 e^{-t} \mathbf{j}-4 e^{-2 t} \mathbf{k}\right)$$

Find the points (if they exist) at which the following planes and curves intersect. $$y=1 ; \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$

The function \(f(x)=\sin n x,\) where \(n\) is a positive real number, has a local maximum at \(x=\pi /(2 n)\) Compute the curvature \(\kappa\) of \(f\) at this point. How does \(\kappa\) vary (if at all) as \(n\) varies?

A baseball leaves the hand of a pitcher 6 vertical feet above home plate and \(60 \mathrm{ft}\) from home plate. Assume that the coordinate axes are oriented as shown in the figure. a. In the absence of all forces except gravity, assume that a pitch is thrown with an initial velocity of \(\langle 130,0,-3\rangle \mathrm{ft} / \mathrm{s}\) (about \(90 \mathrm{mi} / \mathrm{hr}\) ). How far above the ground is the ball when it crosses home plate and how long does it take for the pitch to arrive? b. What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly \(3 \mathrm{ft}\) above the ground? c. A simple model to describe the curve of a baseball assumes that the spin of the ball produces a constant sideways acceleration (in the \(y\) -direction) of \(c \mathrm{ft} / \mathrm{s}^{2}\). Assume a pitcher throws a curve ball with \(c=8 \mathrm{ft} / \mathrm{s}^{2}\) (one-fourth the acceleration of gravity). How far does the ball move in the \(y\) -direction by the time it reaches home plate, assuming an initial velocity of (130,0,-3) ft/s? d. In part (c), does the ball curve more in the first half of its trip to the plate or in the second half? How does this fact affect the batter? e. Suppose the pitcher releases the ball from an initial position of (0,-3,6) with initial velocity \((130,0,-3) .\) What value of the spin parameter \(c\) is needed to put the ball over home plate passing through the point (60,0,3)\(?\)
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