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Chapter 11: Problem 46

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\tan t \mathbf{i}+\left(t+\frac{1}{t}\right) \mathbf{j}-\ln (t+1) \mathbf{k}$$

Short Answer

Expert verified
Question: Find the second and third order derivatives of the vector function \(\mathbf{r}(t) = (\tan t) \mathbf{i} + (t + \frac{1}{t})\mathbf{j} + (-\ln(t + 1))\mathbf{k}\). Answer: The second and third order derivatives of the given vector function are: \(\mathbf{r}^{\prime \prime}(t) = (2\sec^2 t \tan t) \mathbf{i} + (2t^{-3})\mathbf{j} + \frac{1}{(t+1)^2}\mathbf{k}\) and \(\mathbf{r}^{\prime \prime \prime}(t) = (4\sec^2 t(\tan^2t+1)) \mathbf{i} + (-6t^{-4})\mathbf{j} + \frac{-2}{(t+1)^3}\mathbf{k}\).

Step by step solution

01

Derive \(\mathbf{r}^{\prime}(t)\) for each component

To find the first derivative, compute the derivative with respect to \(t\) for each component of the vector separately: $$\mathbf{r}^{\prime}(t) = \frac{d}{dt}(\tan t \mathbf{i}) + \frac{d}{dt}(t+\frac{1}{t})\mathbf{j} + \frac{d}{dt}(-\ln(t+1))\mathbf{k}$$
02

Compute the derivatives for each component in \(\mathbf{r}^{\prime}(t)\)

Calculate the derivatives of each component: $$\mathbf{r}^{\prime}(t) = (\sec^2 t) \mathbf{i} + (1-\frac{1}{t^2})\mathbf{j} + \frac{-1}{t+1}\mathbf{k}$$
03

Derive \(\mathbf{r}^{\prime \prime}(t)\) for each component

Compute the second derivative with respect to \(t\) for each component of the vector separately: $$\mathbf{r}^{\prime \prime}(t) = \frac{d^2}{dt^2}(\sec^2 t \mathbf{i}) + \frac{d^2}{dt^2}(t-\frac{1}{t})\mathbf{j} + \frac{d^2}{dt^2}(-\frac{1}{t+1})\mathbf{k}$$
04

Compute the derivatives for each component in \(\mathbf{r}^{\prime \prime}(t)\)

Calculate the derivatives of each component: $$\mathbf{r}^{\prime \prime}(t) = (2\sec^2 t \tan t) \mathbf{i} + (2t^{-3})\mathbf{j} + \frac{1}{(t+1)^2}\mathbf{k}$$
05

Derive \(\mathbf{r}^{\prime \prime \prime }(t)\) for each component

Compute the third derivative with respect to \(t\) for each component of the vector separately: $$\mathbf{r}^{\prime \prime \prime}(t) = \frac{d^3}{dt^3}(2\sec^2 t \tan t \mathbf{i}) + \frac{d^3}{dt^3}(2t^{-3}\mathbf{j}) + \frac{d^3}{dt^3}(\frac{1}{(t+1)^2}\mathbf{k})$$
06

Compute the derivatives for each component in \(\mathbf{r}^{\prime \prime \prime}(t)\)

Calculate the derivatives of each component: $$\mathbf{r}^{\prime \prime \prime}(t) = (4\sec^2 t(\tan^2t+1)) \mathbf{i} + (-6t^{-4})\mathbf{j} + \frac{-2}{(t+1)^3}\mathbf{k}$$ So, the second and third order derivatives of the given vector function are: $$\mathbf{r}^{\prime \prime}(t) = (2\sec^2 t \tan t) \mathbf{i} + (2t^{-3})\mathbf{j} + \frac{1}{(t+1)^2}\mathbf{k}$$ $$\mathbf{r}^{\prime \prime \prime}(t) = (4\sec^2 t(\tan^2t+1)) \mathbf{i} + (-6t^{-4})\mathbf{j} + \frac{-2}{(t+1)^3}\mathbf{k}$$

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Most popular questions from this chapter

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$

Evaluate the following definite integrals. $$\int_{1 / 2}^{1}\left(\frac{3}{1+2 t} \mathbf{i}-\pi \csc ^{2}\left(\frac{\pi}{2} t\right) \mathbf{k}\right) d t$$

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle 1,2 t, 3 t^{2}\right\rangle ; \mathbf{r}(1)=\langle 4,3,-5\rangle$$

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. Distributive properties a. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+2 \mathbf{u} \cdot \mathbf{v}+|\mathbf{v}|^{2}\) b. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+|\mathbf{v}|^{2}\) if \(\mathbf{u}\) is orthogonal to \(\mathbf{v}\) c. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v})=|\mathbf{u}|^{2}-|\mathbf{v}|^{2}\)

Let \(\mathbf{u}(t)=\left\langle 1, t, t^{2}\right\rangle, \mathbf{v}(t)=\left\langle t^{2},-2 t, 1\right\rangle\) and \(g(t)=2 \sqrt{t}\). Compute the derivatives of the following functions. $$g(t) \mathbf{v}(t)$$
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