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Chapter 10: Problem 28

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{1}{t-1} \mathbf{i}+3 t \mathbf{j} $$

Short Answer

Expert verified
The curve is smooth on the open intervals \( (-\infty, 1) \) and \( (1, \infty) \).

Step by step solution

01

Differentiate

First, differentiate the vector function \( \mathbf{r}(t) \). The derivative of \( \mathbf{r}(t) \) is given by \( \mathbf{r}'(t) \). This gives \( \mathbf{r}'(t)= \left( \frac{d}{dt} \frac{1}{t-1}, \frac{d}{dt} 3t \right) = \left( -\frac{1}{(t-1)^2}, 3 \right) \).
02

Find Where the Derivative is Undefined

Now we need to find when this derivative is undefined. A curve is not differentiable where its derivative is undefined. By looking at the function \( \mathbf{r}'(t)= \left( -\frac{1}{(t-1)^2}, 3 \right) \), we can see that the derivative is undefined when \( t=1 \), because division by zero is undefined.
03

Identify the Open Intervals

Since the only point where the function is not differentiable is \( t=1 \), we can break the function's domain into two open intervals that do not include \( t=1 \). These intervals are all the values of t less than 1 and greater than 1. In other words, the function is smooth on the open intervals \( (-\infty, 1) \) and \( (1, \infty) \).

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Most popular questions from this chapter

In Exercises 39 and \(40,\) find the angle \(\theta\) between \(r(t)\) and \(r^{\prime}(t)\) as a function of \(t .\) Use a graphing utility to graph \(\theta(t) .\) Use the graph to find any extrema of the function. Find any values of \(t\) at which the vectors are orthogonal. $$ \mathbf{r}(t)=3 \sin t \mathbf{i}+4 \cos t \mathbf{j} $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(\theta)=(\theta+\sin \theta) \mathbf{i}+(1-\cos \theta) \mathbf{j} $$

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. $$ \mathbf{r}(t)=\left\langle e^{-t}, e^{t}\right\rangle,(1,1) $$

Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=-\cos t \mathbf{i}-\sin t \mathbf{j} \\ \mathbf{v}(0)=\mathbf{j}+\mathbf{k}, \quad \mathbf{r}(0)=\mathbf{i} \end{array} $$

Evaluate the definite integral. $$ \int_{0}^{\pi / 4}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t $$
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