The indicated figure shows the path of a particle modeled by the vector-valued function

$$r(t)=\langle\pi t-\sin \pi t, 1-\cos \pi t\rangle .$$

The figure also shows the vectors $v(t) /\|v(t)\|$ and $a(t) /\|a(t)\|$ at the indicated values of t.

Find $a_{T}$ and $a_{N}$ at $t=\frac{1}{2}, t=1$, and $t=\frac{3}{2}$.

Angel and Jayden were at track practice. The track is 25 kilometers around. Angel ran 1 lap in 2 minutes. Jayden ran 3 laps in 5 minutes. Who is running faster? Explain your reasoning.

What is the common ratio of the geometric series with the sum

$$\sum_{n=0}^{\infty} 5\left(\frac{4}{5}\right)^{n}?$$

A beam is supported and loaded as shown in the figure. Find the reactions, maximum shear, and maximum moment for the data given in the row(s) assigned from

$$\begin{array}{ccccc} \hline Row & l \mathrm{(in)} & a\mathrm{(in)} & b\mathrm{(in)} & P(\mathrm{l b}) or p(\mathrm{lb} / \mathrm{in}) \\ \hline a & 20 & 16 & 18 & 1000 \\ b & 12 & 2 & 7 & 500 \\ c & 14 & 4 & 12 & 750 \\ d & 8 & 4 & 8 & 1000 \\ e & 17 & 6 & 12 & 1500 \\f & 24 & 16 & 22 & 750 \\ \hline \end{array}$$

Find the point where the graph of $f$ crosses its horizontal asymptote. Sketch the graph of $f$.

$$f(x)=\frac{4 x(x-2)}{(x-3)(x+4)}$$

Solve the formula for the specified variable. Do you recognize the formula? If so, what does it explain? $V=\dfrac{1}{3} B h$ for $B$

Compute the derivatives $F^{\prime}(\mathrm{t})$ and $F^{\prime \prime}(t)$ for the vector-valued function $F(t)=\left(\mathrm{t}, t^2, t^3, t^4\right)$..

Solve the equation for the specified variable.

$N=3 s t^4-5 s v$; for $v$

Let

$$A ( t ) = \left[ \begin{array} { c c } { e ^ { - 3 t } } & { - \sec ^ { 2 } t } \\ { 2 t ^ { 3 } } & { \cos t } \\ { 6 \ln t } & { 36 - 5 t } \end{array} \right] \text { and }B ( t ) = \left[ \begin{array} { c c } { - 7 } & { t ^ { 2 } } \\ { 6 - t } & { 3 t ^ { 3 } + 6 t ^ { 2 } } \\ { 1 + t } & { \cos ( \pi t / 2 ) } \\ { e ^ { t } } & { 1 - t ^ { 3 } } \end{array} \right].$$

Compute the given expression, if possible.

$$B ^ { \prime } ( t ) - e ^ { t } A ( t ).$$

Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any. $f(x)=\frac{x(x-3)}{(x-5)^{2}(x3)^{2}}$

How many eggs are produced as a result of one meiotic series of cell divisions? a. one, b. two, c. three, d. four.

Solve the equation or formula for the specified. $2 g-m=5-g h$, for $g$

Solve the equation for the specified variable.

$S=v t+g t^2$ for $g$.

Write the product using exponents. $(-t) \cdot(-t) \cdot(-t) \cdot(-t) \cdot(-t)$