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Chapter 10: Problem 56

Find \(\mathbf{r}(t)\) for the given conditions. $$ \mathbf{r}^{\prime}(t)=\frac{1}{1+t^{2}} \mathbf{i}+\frac{1}{t^{2}} \mathbf{j}+\frac{1}{t} \mathbf{k}, \quad \mathbf{r}(1)=2 \mathbf{i} $$

Short Answer

Expert verified
\(\mathbf{r}(t) = (\arctan(t) + 2 - \pi/4 )\mathbf{i} - \frac{1}{t}\mathbf{j} + \ln|t|\mathbf{k}\)

Step by step solution

01

Integrate \(\mathbf{r}^{\prime}(t)\) component-wise

The vector \(\mathbf{r}^{\prime}(t)\) consists of three components, and each of these components should be integrated separately. Integrating the given derivative, we get \(\int \frac{1}{1+t^{2}}dt\mathbf{i}+\int \frac{1}{t^{2}}dt \mathbf{j}+\int \frac{1}{t}dt \mathbf{k}\)
02

Evaluate the integrals

The antiderivatives for each of the components are as follows: \( \int \frac{1}{1+t^{2}}dt = \arctan(t)\mathbf{i}+const_i\), \( \int \frac{1}{t^{2}}dt = -\frac{1}{t}\mathbf{j}+const_j\), and \( \int \frac{1}{t}dt=\ln |t|\mathbf{k}+const_k\). Here, \(const_i, const_j, const_k\) are constants of integration.
03

Calculate the constants of integration from \(\mathbf{r}(1)=2 \mathbf{i}\)

We know that the vector function \(\mathbf{r}(t)\) equals \(2 \mathbf{i}\) when \(t=1\). This gives us three equations, one for each component of the vector: For the \(\mathbf{i}\)-component: \(\arctan(1) + const_i = 2\), for the \(\mathbf{j}\)-component: \(\frac{-1}{1} + const_j = 0\), and for the \(\mathbf{k}\)-component: \(\ln(1) + const_k = 0\). Solving these equations gives \(const_i = 2 - \pi/4, const_j = 1,\) and \(const_k = 0\).
04

Incorporate the constants of integration into \(\mathbf{r}(t)\)

Finally, we can obtain \(\mathbf{r}(t)\) by putting the constants of integration from step 3 into the integral-like function from step 2. The result is \(\mathbf{r}(t) = (\arctan(t) + 2 - \pi/4 )\mathbf{i} - \frac{1}{t}\mathbf{j} + \ln|t|\mathbf{k}\).

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Most popular questions from this chapter

In Exercises \(27-34,\) find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j} $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=(t-1) \mathbf{i}+\frac{1}{t} \mathbf{j}-t^{2} \mathbf{k} $$

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. $$ \mathbf{r}(t)=3 t \mathbf{i}+(t-1) \mathbf{j},(3,0) $$

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. $$ \mathbf{r}(t)=3 \sin t \mathbf{i}+4 \cos t \mathbf{j} $$
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