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Chapter 10: Problem 34

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+3 t \mathbf{k} $$

Short Answer

Expert verified
The curve given by the vector-valued function \(\mathbf{r}(t)=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+3 t \mathbf{k}\) is smooth for all real numbers.

Step by step solution

01

Differentiating the vector function

The derivative of \(\mathbf{r}(t)\) can be found by differentiating each component of the function separately. The derivative of \(e^{t}\) is \(e^{t}\), the derivative of \(-e^{-t}\) is \(e^{-t}\) and the derivative of \(3t\) is \(3\). Thus, the derivative, \(\mathbf{r}'(t)\), is given by \(\mathbf{r}'(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j}+3 \mathbf{k}\)
02

Analyzing the continuity of the derivative

The function \(\mathbf{r}'(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j}+3 \mathbf{k}\) is continuous for all real numbers since the exponential function and the constant function are both continuous for all real numbers. Therefore, the curve is smooth for all real numbers.

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Most popular questions from this chapter

Use the definition of the derivative to find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=\langle 0, \sin t, 4 t\rangle $$

The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=\langle t-\sin t, 1-\cos t\rangle,(\pi, 2) $$

Consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(\mathbf{r}(t)=b(\omega t-\sin \omega t) \mathbf{i}+b(1-\cos \omega t) \mathbf{j}\) where \(\omega\) is the constant angular velocity of the circle and \(b\) is the radius of the circle. Find the maximum speed of a point on the circumference of an automobile tire of radius 1 foot when the automobile is traveling at 55 miles per hour. Compare this speed with the speed of the automobile.

Prove the property. In each case, assume that \(\mathbf{r}, \mathbf{u},\) and \(\mathbf{v}\) are differentiable vector-valued functions of \(t,\) \(f\) is a differentiable real-valued function of \(t,\) and \(c\) is a scalar. $$ D_{t}\left[\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)\right]=\mathbf{r}(t) \times \mathbf{r}^{\prime \prime}(t) $$

Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ $$ \text { Find the velocity vector and show that it is orthogonal to } \mathbf{r}(t) $$
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