91影视

$r\left(t\right)=\left(e^t\sin t,e^t\cos t,e^t\right)$

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.

$\mathbf{r}\left(t\right)=鉄�3t-4,-2t+5,t+3鉄�,[1,5]$

In Exercises 21鈥�23 you are given a vector function $r$and a scalar function $t=f(蟿)$. Compute$\frac{d\mathbf{r}}{d蟿}$in the following two ways:

(a) By using the chain rule $\frac{d\mathbf{r}}{d蟿}=\frac{d\mathbf{r}}{dt}\frac{dt}{d蟿}$.

(b) By substituting $t=f(蟿)$into the formula for $r$. Ensure that your two answers are consistent.

For each of the vector-valued functions, find the unit tangent vector .

$r\left(t\right)=(t,t^2)$

Given a vector-valued function r(t) with domain $\mathrm{鈩�},$ what is the relationship between the graph of r(t) and the graph of kr(t), where k > 1 is a scalar?

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.

$\mathbf{r}\left(t\right)=鉄�3\cos 4t,3\sin 4t鉄�,\left[0,\frac{蟺}{2}\right]$

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.

$\mathbf{r}\left(t\right)=(t,2t\sin t,2t\cos t),t=蟺$

In Exercises 21鈥�23 you are given a vector function $r$and a scalar function $t=f(蟿)$. Compute $\frac{d\mathbf{r}}{d蟿}$in the following two ways:

(a) By using the chain rule $\frac{d\mathbf{r}}{d蟿}=\frac{d\mathbf{r}}{dt}\frac{dt}{d蟿}$.

(b) By substituting $t=f(蟿)$into the formula for $r$. Ensure that your two answers are consistent.

For each of the vector-valued functions, find the unit tangent vector.

$r\left(t^2\right)=(t^2+5,5t,4t^3)$

Given a vector-valued function r(t) with domain $\mathrm{鈩�},$what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?