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).

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 fixed-end column with circular cross section is acted on by compressive axial load $P$. The $18 -\text{ft}$-long-column has an outer diameter of $5\ \text{in.}$, a thickness of $0.5\ \text{in.}$, and is made of aluminum with a modulus of elasticity of $10,000\ \mathrm{ksi}$. Find the buckling load of the column.

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

Calculate $\frac{d y}{d x}$. You need not expand your answers. $y=\frac{3 x-1}{(x-5)(x-4)(x-1)}$

The tee shape shown in Figure P8.64b/65b is used as a short post to support a load of $P=4,600 \mathrm{lb}$. The load $P$ is applied at a distance of 5 in. from the surface of the flange, as shown in Figure P8.64a/65a. Determine the normal stresses at points $H$ and $K$, which are located on section $a-a$.

A uniform rope of cross-sectional area 0.50 cm² breaks when the tensile stress in it reaches

$$6.00 × 10^6 N/m^2.$$

(a) What is the maximum load that can be lifted slowly at a constant speed by the rope? (b) What is the maximum load that can be lifted by the rope with an acceleration of 4.00 m/s²?

What is the sequence of buttons you need to press to enter the number $5.8 \times 10^{-8}$ into a scientific calculator?

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 $x^{2} \cos \left(\frac{k \pi x}{l}\right)$.

A steel control rod is 5.5 ft long and must not stretch more than 0.04 in. when a 2-kip tensile load is applied to it. Knowing that $E=29 \times 106$ psi, determine (a) the smallest diameter rod that should be used, (b) the corresponding normal stress caused by the load.

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

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

There are 1,012 souvenir paperweights that need to be packed in boxes. Each box will hold 12 paperweights. How many boxes will be needed?

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

Use one of the formulas for special series to find the sum of the series.

$$\sum_{i=1}^{42} 1$$