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Chapter 11: Problem 14

Differentiate the following functions. $$\mathbf{r}(t)=\left\langle(t+1)^{-1}, \tan ^{-1} t, \ln (t+1)\right\rangle$$

Short Answer

Expert verified
Answer: The derivative of the vector function is given by \(\frac{d\mathbf{r}}{dt} = \left\langle -(t+1)^{-2}, \frac{1}{1 + t^2}, \frac{1}{(t+1)} \right\rangle\).

Step by step solution

01

Differentiate the first component \((t+1)^{-1}\)

To differentiate \((t+1)^{-1}\) with respect to \(t\), use the power rule: $$\frac{d}{dt}\left((t+1)^{-1}\right) = -1\times(t+1)^{-2}$$
02

Differentiate the second component \(\tan ^{-1} t\)

To differentiate \(\tan ^{-1} t\) with respect to \(t\), use the formula for the derivative of the inverse tangent function: $$\frac{d}{dt}\left(\tan ^{-1} t\right) = \frac{1}{1 + t^2}$$
03

Differentiate the third component \(\ln (t+1)\)

To differentiate \(\ln (t+1)\) with respect to \(t\), use the chain rule: $$\frac{d}{dt}\left(\ln (t+1)\right) = \frac{1}{(t+1)}$$
04

Combine the derivatives to find the derivative of the whole vector function

Now that we have the derivatives of each component, we can put them together to find the derivative of the whole vector function \(\mathbf{r}(t)\): $$\frac{d\mathbf{r}}{dt} = \left\langle \frac{d}{dt}\left((t+1)^{-1}\right), \frac{d}{dt}\left(\tan ^{-1} t\right), \frac{d}{dt}\left(\ln (t+1)\right) \right\rangle$$ Substitute the derivatives we found for each component: $$\frac{d\mathbf{r}}{dt} = \left\langle -1\times(t+1)^{-2}, \frac{1}{1 + t^2}, \frac{1}{(t+1)} \right\rangle$$

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Most popular questions from this chapter

A pair of nonzero vectors in the plane is linearly dependent if one vector is a scalar multiple of the other. Otherwise, the pair is linearly independent. a. Which pairs of the following vectors are linearly dependent and which are linearly independent: \(\mathbf{u}=\langle 2,-3\rangle\) \(\mathbf{v}=\langle-12,18\rangle,\) and \(\mathbf{w}=\langle 4,6\rangle ?\) b. Geometrically, what does it mean for a pair of nonzero vectors in the plane to be linearly dependent? Linearly independent? c. Prove that if a pair of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is linearly independent, then given any vector \(w\), there are constants \(c_{1}\) and \(c_{2}\) such that \(\mathbf{w}=c_{1} \mathbf{u}+c_{2} \mathbf{v}\)

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle 3 t^{12}-t^{2}, t^{8}+t^{3}, t^{-4}-2\right\rangle$$

Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0} .\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\langle\sqrt{2 t+1}, \sin \pi t, 4\rangle ; t_{0}=4$$

Let $$\mathbf{u}(t)=2 t^{3} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}-8 \mathbf{k} \text { and } \mathbf{v}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-e^{2 t} \mathbf{k}$$ Compute the derivative of the following functions. $$\mathbf{u}(t) \cdot \mathbf{v}(t)$$

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\langle 0,2,2 t\rangle ; \mathbf{r}(1)=\langle 4,3,-5\rangle$$
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