Determine whether r(t) is continuous at t=0. Explain your reasoning.
r(t)=3 sin ti-2tj

Determine whether r(t) is continuous at t=0. Explain your reasoning.

$$\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + \frac { 1 } { t } \mathbf { j } + t \mathbf { k }$$

Let r(t)=cos t i+sin t j+k. Find

$$\lim _ { t \rightarrow 0 } \left( \mathbf { r } ( t ) - \mathbf { r } ^ { \prime } ( t ) \right)$$

Let

$$\mathbf { r } ( t ) = \left\langle \cos t , \sin t , t ^ { 3 / 2 } \right\rangle$$

. Find

$$\frac { d s } { d t }$$

Find a change of parameter

$$t = g ( \tau )$$

for the semicircle

$$\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } \quad ( 0 \leq t \leq \pi )$$

such that the semicircle is traced clockwise as

$$\tau$$

varies over the interval [0, 1].

Let

$$\mathbf { r } ( t ) = \left\langle \cos t , \sin t , t ^ { 3 / 2 } \right\rangle$$

. Find

$$\left\| \mathbf { r } ^ { \prime } ( t ) \right\|$$

Let

$$\mathbf { r } ( t ) = \left\langle \cos t , \sin t , t ^ { 3 / 2 } \right\rangle$$

. Find

$$\frac { d s } { d t }$$

A particle moves along the plane curve C described by r(t) = ti + t²j. (b) Find the curvature K of the plane curve at t = 0, t = 1, and t = 2.

The controversy generated by passage of the Force Bill reflected ongoing debates in the United States over a. the power of the presidency. b. federalism and states' rights. c. the legal use of military force. d. First Amendment rights of assembly and protest.

These exercises are concerned with the problem of creating a single smooth curve by piecing together two separate smooth curves. If two smooth curves

$$C_1$$

and

$$C_2$$

are joined at a point P to form a curve C, then we will say that

$$C_1$$

and

$$C_2$$

make a smooth transition at P if the curvature of C is continuous at P. Show that for the curve

$$y = x ^ { 2 }$$

for

$$x<0$$

,

$$y=x^4$$

for

$$x \geq 0$$

the transition at x=0 is not smooth.

In each part, use the formulas $\kappa ( t ) = \frac { \left| x ^ { \prime } y ^ { \prime \prime } - y ^ { \prime } x ^ { \prime \prime \prime } \right| } { \left( x ^ { \prime 2 } + y ^ { \prime 2 } \right) ^ { 3 / 2 } }$ and $\kappa ( x ) = \frac { \left| d ^ { 2 } y / d x ^ { 2 } \right| } { \left[ 1 + ( d y / d x ) ^ { 2 } \right] ^ { 3 / 2 } }$, to help find the radius of curvature at the stated points. Then sketch the graph together with the osculating circles at those points.

$$x = 2 \cos t , y = \sin t \quad ( 0 \leq t \leq 2 \pi ) \ at \ t = 0 \ and \\\ t = \pi / 2$$

How to describe settings of 1984 novel?

A particle moves along the plane curve C described by r(t) = ti + t²j. (a) Find the length of C on the interval 0 ≤ t ≤ 2.

If C is the graph of a smooth vector-valued function r(t), then the unit tangent, unit normal, and binormal to C at t are defined, respectively, by T(t)=, N(t)=, B(t)=___