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Chapter 11: Problem 11

Differentiate the following functions. $$\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-4 e^{2 t} \mathbf{k}$$

Short Answer

Expert verified
The derivative of the given vector function with respect to \(t\) is \(\mathbf{r'}(t) = e^{t} \mathbf{i} - 2 e^{-t} \mathbf{j} - 8 e^{2t} \mathbf{k}\).

Step by step solution

01

Differentiate the i-component #

First, we will differentiate the i-component of \(\mathbf{r}(t)\) with respect to \(t\). So, we need to find the derivative of \(e^t\) with respect to \(t\). By using the exponential rule for differentiation, we get the derivative of the i-component: $$\frac{d}{dt} e^t = e^t$$.
02

Differentiate the j-component #

Next, we will differentiate the j-component of \(\mathbf{r}(t)\) with respect to \(t\). So, we need to find the derivative of \(2 e^{-t}\) with respect to \(t\). By using the chain rule, we get the derivative of the j-component: $$\frac{d}{dt} (2 e^{-t}) = -2 e^{-t}$$.
03

Differentiate the k-component #

Lastly, we will differentiate the k-component of \(\mathbf{r}(t)\) with respect to \(t\). So, we need to find the derivative of \(-4 e^{2t}\) with respect to \(t\). By using the chain rule, we get the derivative of the k-component: $$\frac{d}{dt} (-4 e^{2t}) = -8 e^{2t}$$.
04

Combine the derivatives #

Now we will combine the derivatives of the i, j, and k components to form the derivative of \(\mathbf{r}(t)\). The derivative of \(\mathbf{r}(t)\) with respect to \(t\) is given by: $$\mathbf{r'}(t) = \frac{d\mathbf{r}}{dt} = e^{t} \mathbf{i} - 2 e^{-t} \mathbf{j} - 8 e^{2t} \mathbf{k}$$. Thus, the derivative of \(\mathbf{r}(t) = e^t \mathbf{i} + 2 e^{-t} \mathbf{j} - 4 e^{2t} \mathbf{k}\) with respect to \(t\) is $$\mathbf{r'}(t) = e^{t} \mathbf{i} - 2 e^{-t} \mathbf{j} - 8 e^{2t} \mathbf{k}$$.

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Most popular questions from this chapter

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. $$\mathbf{u}(t) \cdot \mathbf{v}(t)$$

Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{t} \mathbf{i}+e^{t} \cos \left(\pi e^{t}\right) \mathbf{j}\right) d t$$

Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{-t} \mathbf{i}+2 e^{2 t} \mathbf{j}-4 e^{t} \mathbf{k}\right) d t$$

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$$

Evaluate the following definite integrals. $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t$$
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