Explain why the following statement does not need to be justified. “The midpoint of a segment is a point on the segment that divides it into two congruent segments.”

Decide whether each statement is always true, sometimes true, or never true. Justify your answers. a) Two line segments with the same midpoint have the same length. b) Two parallel line segments have the same midpoint. c) The midpoint of a line segment is the only point that divides it into two equal parts. d) A point equidistant from the endpoints of a line segment is the midpoint of the line segment.

Each series satisfies the hypotheses of the alternating series test. Approximate the sum of the series to two decimal-place accuracy.

$$1 - \frac { 1 } { 2 ! } + \frac { 1 } { 4 ! } - \frac { 1 } { 6 ! } + \cdots$$

Use the geometric series

$$f ( x ) = \frac { 1 } { 1 - x } = \sum _ { k = 0 } ^ { \infty } x ^ { k }$$

, for |x|<1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.

Let $G$ be an Abelian group and $H = \{ x \in G | | x |$ is 1 or even $\} .$ Give an example to show that $H$ need not be a subgroup of $G .$

A propped cantilever beam is subjected to uniform load $q$. The beam has flexural rigidity $E I=2000\ \mathrm{kip}-\mathrm{ft}^2$ and the length of the beam is $10 \mathrm{ft}$. Find the intensity $q$ of the distributed load if the maximum displacement of the beam is $\delta_{\max }=0.125\ \mathrm{in}$.

Suppose that you want to find the Fourier series of $f(x)=x+x^{2}$. Explain why to find $b_k$ you would need only to integrate $x \sin \left(\frac{k \pi x}{l}\right)$ and to find $a_k$ you would need only to integrate $x^{2} \cos \left(\frac{k \pi x}{l}\right)$.

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $g(x)=\frac{x^{3}}{x^{2}-4}$

If you want to tile the kitchen with $12" \times 12"$ tiles, how many tiles will you need to buy?

(a) Based on Tiresias' prediction, which heroic qualities will Odysseus need to rely upon as he continues his journey? Explain.

A cast-iron tube is used to support a compressive load. Knowing that $E=69 \mathrm{GPa}$ and that the maximum allowable change in length is $0.025 \%$. Actermine $(a)$ the maximum normal stress in the tube, $(b)$ the minimum wall thickness for a load of $7.2 \mathrm{kN}$ if the outside diameter of the tube is $50 \mathrm{~mm}$.

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $g(x)=2 x^{3}+x^{2}+1$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $f(x)=\frac{x^{2}-2}{x^{2}-4}$

a) Determine the load impedance for the circuit shown in Fig. that will result in maximum average power being transferred to the load if $\omega=5 \mathrm{krad} / \mathrm{s}$. b) Determine the maximum average power delivered to the load from part (a) if $v_g=$ $80 \cos 5000 t \mathrm{~V}$. c) Repeat part (a) when $Z_{\mathrm{L}}$. consists of two components from Appendix $\mathrm{H}$ whose values yield a maximum average power closest to the value calculated in part (b).