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 $\delta_{\max }=0.125\ \mathrm{in}$.

Use a graphing utility to graph the power series. The series represents a well-known function. Identify the function from the graph of its power series.

$$f(x)=\sum_{n=0}^{\infty}(-1)^n x^n, \quad-1<x<1$$

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 $50 \mathrm{~mm}$.

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 $a-a$.

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

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.

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

What factor would you need to multiply by $(3d + 1)$ to get $9 d^{2}+6 d+1 ?$

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

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.

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.) $f(x)=\frac{x^{2}-2}{x^{2}-4}$

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.”

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?