Evaluate the expression. Give your answer in both radians and degrees.

$$cos^{-1}0$$

In a cathode ray tube, electrons initially at rest are accelerated by a uniform electric field of magnitude $4.0 \times 10 ^ { 5 } \mathrm { N } / \mathrm { C }$ during the first 5.0 cm of the tube's length; then they move at essentially constant velocity another 45 cm before hitting the screen. (a) Find the speed of the electrons when they hit the screen. (b) How long does it take them to travel the length of the tube?

Graph each equation in a

$$\left[ - 2 \pi , 2 \pi , \frac { \pi } { 2 } \right]$$

by [-3, 3, 1] viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent.

$$y = \frac { 2 \tan \frac { x } { 2 } } { 1 + \tan ^ { 2 } \frac { x } { 2 } }$$

(a) Give a counterexample to show that $$ ( A B ) ^ { - 1 } \neqA ^ { - 1 } B ^ { - 1 } $$ in general. (h) Under what conditions on A and B is $$ ( A B ) ^ { - 1 } =A ^ { - 1 } B ^ { - 1 } ? $$ Prove your assertion.

Suppose $A$ and $B$ are sets, and $f$ and $g$ are functions with $f:A\rightarrow B$ and $g:B\rightarrow A$. Prove: If $g\circ f=\text{id}_A$ and $f\circ g=\text{id}_B$, then $f$ is invertible and $g=f^{-1}$. Note: This result is a converse to Proposition 26.9.

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to

$$ℤ_p × ℤ_p$$

for some prime p.

Determine the decimal value of each signed binary number in the 2’s complement form: (a) 10011001 (b) 01110100 (c) 10111111

The length of a rectangle is 2 inches more than the width w. What expression in terms of w could you use for the length of the rectangle?

A 1-m^3 tank contains 2.841 kg of steam at 0.6 MPa. Determine the temperature of the steam, using (a) the ideal-gas equation, (b) the van der Waals equation, and (c) the steam tables. Answers: (a) 457.6 K, (b) 465.9 K, (c) 473 K

Solve each system for x and y expressing either value in terms of a or b, if necessary. Assume that

$$a \neq 0 \text { and } b \neq 0.$$

For the linear function

$$f ( x ) = m x + b , f ( - 2 ) = 11 \text { and } f ( 3 ) = - 9. Find m and b.$$

Does ( 4 , - 1 ) satisfy x - 2 y = 6 ?

Use expanded forms to convert the given base two numeral to base ten.

$$10101_{ t w o}$$

$$\left. \begin{array} { l } { \text { What are the orders of the elements of } D _ { 15 } ? \text { How many elements } } \\ { \text { have each of these orders? } } \end{array} \right.$$

Sketch the graph of each equation.

$$y = \frac { 3 } { 4 } x - 1$$