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Chapter 12: Problem 7

Differentiate the following functions. $$\mathbf{r}(t)=\left\langle\cos t, t^{2}, \sin t\right\rangle$$

Short Answer

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Question: Find the derivative of the vector-valued function \(\mathbf{r}(t)=\left\langle\cos t, t^{2}, \sin t\right\rangle\) with respect to \(t\). Answer: The derivative of the vector-valued function \(\mathbf{r}(t)\) with respect to \(t\) is \(\frac{d\mathbf{r}}{dt} = \left\langle -\sin t, 2t, \cos t\right\rangle\).

Step by step solution

01

Identify the components of the vector function

The vector-valued function is given as: $$\mathbf{r}(t)=\left\langle\cos t, t^{2}, \sin t\right\rangle$$ The components of the vector \(\mathbf{r}(t)\) are: \(x(t) = \cos t\), \(y(t) = t^2\), and \(z(t) = \sin t\).
02

Calculate the derivative of each component

To find the derivative of \(x(t)\), \(y(t)\), and \(z(t)\) with respect to \(t\), we use the following rules of differentiation: 1. The derivative of \(\cos t\) with respect to \(t\) is \(-\sin t\). 2. The derivative of \(t^2\) with respect to \(t\) is \(2t\). 3. The derivative of \(\sin t\) with respect to \(t\) is \(\cos t\). So, we find: $$\frac{dx}{dt}=-\sin t$$ $$\frac{dy}{dt}=2t$$ $$\frac{dz}{dt}=\cos t$$
03

Create a vector from the component derivatives

Now we will assemble the derivatives into a vector, which represents the derivative of the vector-valued function \(\mathbf{r}(t)\): $$\frac{d\mathbf{r}}{dt} = \left\langle\frac{dx}{dt}, \frac{dy}{dt},\frac{dz}{dt}\right\rangle$$ Substitute the calculated derivatives into the above expression: $$\frac{d\mathbf{r}}{dt} = \left\langle -\sin t, 2t, \cos t\right\rangle$$ Thus, the derivative of the given vector-valued function is: $$\frac{d\mathbf{r}}{dt} = \left\langle -\sin t, 2t, \cos t\right\rangle$$

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Most popular questions from this chapter

Carry out the following steps to determine the (smallest) distance between the point \(P\) and the line \(\ell\) through the origin. a. Find any vector \(\mathbf{v}\) in the direction of \(\ell\) b. Find the position vector u corresponding to \(P\). c. Find \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). d. Show that \(\mathbf{w}=\mathbf{u}-\) projy \(\mathbf{u}\) is a vector orthogonal to \(\mathbf{v}\) whose length is the distance between \(P\) and the line \(\ell\) e. Find \(\mathbf{w}\) and \(|\mathbf{w}| .\) Explain why \(|\mathbf{w}|\) is the distance between \(P\) and \(\ell\). \(P(1,1,-1) ; \ell\) has the direction of $$\langle-6,8,3\rangle$$.

Prove that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) is $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$

Find the domains of the following vector-valued functions. $$\mathbf{r}(t)=\sqrt{t+2} \mathbf{i}+\sqrt{2-t} \mathbf{j}$$

An object moves along an ellipse given by the function \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle,\) for \(0 \leq t \leq 2 \pi,\) where \(a > 0\) and \(b > 0\) a. Find the velocity and speed of the object in terms of \(a\) and \(b\) for \(0 \leq t \leq 2 \pi\) b. With \(a=1\) and \(b=6,\) graph the speed function, for \(0 \leq t \leq 2 \pi .\) Mark the points on the trajectory at which the speed is a minimum and a maximum. c. Is it true that the object speeds up along the flattest (straightest) parts of the trajectory and slows down where the curves are sharpest? d. For general \(a\) and \(b\), find the ratio of the maximum speed to the minimum speed on the ellipse (in terms of \(a\) and \(b\) ).

Trajectory with a sloped landing Assume an object is launched from the origin with an initial speed \(\left|\mathbf{v}_{0}\right|\) at an angle \(\alpha\) to the horizontal, where \(0 < \alpha < \frac{\pi}{2}\) a. Find the time of flight, range, and maximum height (relative to the launch point) of the trajectory if the ground slopes downward at a constant angle of \(\theta\) from the launch site, where \(0 < \theta < \frac{\pi}{2}\) b. Find the time of flight, range, and maximum height of the trajectory if the ground slopes upward at a constant angle of \(\theta\) from the launch site.
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