A squirrel has $x$- and $y$-coordinates $(1.1 \mathrm{~m}, 3.4 \mathrm{~m})$ at time $t_1=0$ and coordinates $(5.3 \mathrm{~m},-0.5 \mathrm{~m})$ at time $t_2=3.0 \mathrm{~s}$. For this time interval, find
(b) the magnitude and direction of the average velocity.

Let $\varphi$ : G $\rightarrow$ GL(V) be a representation of G. The centralizer of $\varphi$ is defined to be the set of all linear transformations, A, from V to itself such that

$$A \varphi(g)=\varphi(g) A$$

for all g $\in$ G (i.e., the linear transformations of V which commute with all $\varphi(g)^{\prime}s$). (a) Prove that a linear transformation A from V to V is in the centralizer of $\varphi$ if and only if it is an FG-module homomorphism from V to itself (so the centralizer of $\varphi$ is the same as the $\left.\operatorname{ring} \operatorname{Hom}_{F G}(V, V)\right)$. (b) Show that if z is in the center of G then $\varphi(z)$ is in the centralizer of $\varphi$. (c) Assume $\varphi$ is an irreducible representation (so V is a simple FG-module). Prove that if H is any finite abelian subgroup of GL(V) such that A$\varphi$(g) = $\varphi$(g)A for all A $\in$ H then H is cyclic (in other words, any finite abelian subgroup of the multiplicative group of units in the ring Hom$_{F G}$(V, V) is cyclic). Hom$_{F G}$(V, V) is a division ring, so this reduces to proving that a finite abelian subgroup of the multiplicative group of nonzero elements in a division ring is cyclic. Show that the division subring generated by an abelian subgroup of any division ring is a field and use Proposition 18. (d) Show that if $\varphi$ is a faithful irreducible representation then the center of G is cyclic. (e) Deduce from (d) that if G is abelian and $\varphi$ is any irreducible representation then G/ker $\varphi$ is cyclic.

A child's hoop of mass M and radius b rolls in a straight line with velocity v. Its top is given a light tap with a stick at right angles to the direction of motion. The impulse of the blow is $I.$ a. Show that this results in a deflection of the line of rolling by angle $\phi=I / M v,$ assuming that the gyroscope approximation holds and neglecting friction with the ground. b. Show that the gyroscope approximation is valid provided $F \ll M v^{2} / b,$ where F is the peak applied force.

The Tethered Satellite discussed in this module is producing 5.00 kV, and a current of 10.0 A flows. (a) What magnetic drag force does this produce if the system is moving at 7.80 km/s? (b) How much kinetic energy is removed from the system in 1.00 h, neglecting any change in altitude or velocity during that time? (c) What is the change in velocity if the mass of the system is 100,000 kg? (d) Discuss the long term consequences (say, a week-long mission) on the space shuttle's orbit, noting what effect a decrease in velocity has and assessing the magnitude of the effect.

In 1975, an airplane carrying an atomic clock flew back and forth for $15 \mathrm{~h}$ at an average speed of $140 \mathrm{~m} / \mathrm{s}$ as part of a time-dilation experiment. The time on the clock was compared to the time on an atomic clock kept on the ground. How much time did the airborne clock "lose" with respect to the clock on the ground?

The matrix P that multiplies (x, y, z) to give (z, x, y) is also a rotation matrix. Find P and P³. The rotation axis a = (1, 1, 1) doesn't move, it equals Pa. What is the angle of rotation from v = (2, 3, -5) to Pv = (-5, 2, 3)?

(Printing a String) Write a program that defines and uses macro PRINT to print a string value.

Find the distance between each pair of points. (x+y, y) and (x-y, x)

You are offered a job that pays $30,000 for the first year with an annual increase of 5% per year beginning in the second year. That is, beginning in year 2, your salary will be 1.05 times what it was in the previous year. What can you expect to earn in your sixth year on the job? Round to the nearest dollar.

Differentiate the functions in problem with respect to x. $y=\frac{\sin x}{e^{x}-x}$

Find the exponential decay equation for a population that halves in size every unit of time and that has 1024 individuals at time 0.

Enzymes that are only produced when a substrate is present are termed _____.

Tomás and David are exact opposites. Read the descriptions of Tomás and then write statements about David.

$$\begin{matrix} \text{Tomas es bueno. } & \text{\_\_\_\_\_\_\_\_\_\_\_\_}\\ \end{matrix}$$

An RC series circuit consists of an AC source, a $33.0 - \mu \text F$ capacitor, and a 30.0-$\Omega$ resistor. If the source emf is given by $\varepsilon = \varepsilon_ { max } \sin ( \omega t )$, where $E _ { \max } = 180.0 \mathrm { V }$ and $\omega = 20.0 \mathrm { s } ^ { - 1 }$, what is the current amplitude?