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Chapter 11: Problem 39
Calculate the work done in the following situations. A sled is pulled \(10 \mathrm{m}\) along horizontal ground with a constant force of \(5 \mathrm{N}\) at an angle of \(45^{\circ}\) above the horizontal.
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Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle e^{t}, \sin t, \sec ^{2} t\right\rangle ; \mathbf{r}(0)=\langle 2,2,2\rangle$$
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. $$\mathbf{v}(g(t))$$
Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=t e^{t} \mathbf{i}+t \sin t^{2} \mathbf{j}-\frac{2 t}{\sqrt{t^{2}+4}} \mathbf{k}$$
Evaluate the following definite integrals. $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t$$
Show that the (least) distance \(d\) between a point \(Q\) and a line \(\mathbf{r}=\mathbf{r}_{0}+t \mathbf{v}\) (both in \(\mathbb{R}^{3}\) ) is \(d=\frac{|\overrightarrow{P Q} \times \mathbf{v}|}{|\mathbf{v}|},\) where \(P\) is a point on the line.
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