91Ó°ÊÓ

Compute the derivatives of the vector-valued functions. $$\mathbf{r}(t)=\tan (2 t) \mathbf{i}+\sec (2 t) \mathbf{j}+\sin ^{2}(t) \mathbf{k}$$

Compute the derivatives of the vector-valued functions. $$\mathbf{r}(t)=3 \mathbf{i}+4 \sin (3 t) \mathbf{j}+t \cos (t) \mathbf{k}$$

Compute the derivatives of the vector-valued functions. $$\mathbf{r}(t)=t^{2} \mathbf{i}+t e^{-2 t} \mathbf{j}-5 e^{-4 t} \mathbf{k}$$

For the following problems, find a tangent vector at the indicated value of \(t .\) $$\mathbf{r}(t)=t \mathbf{i}+\sin (2 t) \mathbf{j}+\cos (3 t) \mathbf{k} ; t=\frac{\pi}{3}$$

For the following problems, find a tangent vector at the indicated value of \(t .\) $$\mathbf{r}(t)=3 t^{3} \mathbf{i}+2 t^{2} \mathbf{j}+\frac{1}{t} \mathbf{k} ; t=1$$

For the following problems, find a tangent vector at the indicated value of \(t .\) $$\mathbf{r}(t)=3 e^{t} \mathbf{i}+2 e^{-3 t} \mathbf{j}+4 e^{2 t} \mathbf{k} ; t=\ln (2)$$

For the following problems, find a tangent vector at the indicated value of \(t .\) $$\mathbf{r}(t)=\cos (2 t) \mathbf{i}+2 \sin t \mathbf{j}+t^{2} \mathbf{k} ; t=\frac{\pi}{2}$$

Find the unit tangent vector for the following parameterized curves. $$\mathbf{r}(t)=6 \mathbf{i}+\cos (3 t) \mathbf{j}+3 \sin (4 t) \mathbf{k}, \quad 0 \leq t<2 \pi$$

Find the unit tangent vector for the following parameterized curves. $$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\sin t \mathbf{k}, \quad 0 \leq t<2 \pi$$ Two views of this curve are presented here:

Find the unit tangent vector for the following parameterized curves. $$\mathbf{r}(t)=3 \cos (4 t) \mathbf{i}+3 \sin (4 t) \mathbf{j}+5 t \mathbf{k}, 1 \leq t \leq 2$$