91影视

Find the curvature of each of the vector-valued functions defined in Exercises 39鈥�44.

$\mathit{r}\left(t\right)=\left(t,t^2,t^3\right).$

Evaluate the integrals in Exercises 40鈥�44.

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.
$\left(\right(\sin t)i+(\cos t)j+tk)脳\left(\right(\cos t)i+(\sin t\left)j\right)$

Find the curvature of each of the vector-valued functions defined in Exercises 39鈥�44.

$\mathit{r}\left(t\right)=\left(t\sin t,t\cos t,t\right)$

Using the definitions of the normal plane and rectifying plane in Exercises 20 and 21, respectively, find the equations of these planes at the specified points for the vector functions in Exercises 40鈥�42. Note: These are the same functions as in Exercises 35, 37, and 39.

Evaluate the integrals in Exercises 40鈥�44.

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.

$(1,t,t^2)脳(1,t^2,t^3)$

Evaluate the integrals:

Evaluate the limits in Exercises 42鈥�45.

$\underset{t鈫�0}{lim}(\sin 3t,\cos 3t)$

Using the definitions of the normal plane and rectifying plane in Exercises 20 and 21, respectively, find the equations of these planes at the specified points for the vector functions in Exercises 40鈥�42. Note: These are the same functions as in Exercises 35, 37, and 39.

$\mathbf{r}\left(t\right)=鉄�\sin 鈦�2t,\cos 鈦�2t,t鉄�\text{at}t=\frac{蟺}{2}$