At some point in their lives most people will suffer from at least one onset of low back pain. This disorder can trigger excruciating pain and temporary disability, but its causes are hard to diagnose. It is well known that low back pain alters motor trunk patterns; thus it is of interest to study the causes for these alterations and their extent. Due to the different possible causes of this type of pain, a "control" group of people is hard to obtain for laboratory studies. However, pain can be stimulated in healthy people and muscle movement ranges can be compared. Controlled back pain can be induced by injecting saline solution directly into related muscles or ligaments. The transfer function from infusion rate to pain response was obtained experimentally by injecting a 5% saline solution at six different infusion rates over a period of 12 minutes. Subjects verbally rated their pain every 15 seconds on a scale from 0 to 10, with 0 indicating no pain and 10 unbearable pain. Several trials were averaged and the data was fitted to the following transfer function:

$$G(s)=\frac{9.72 \times 10^{-8}(s+0.0001)}{(s+0.009)^{2}\left(s^{2}+0.018 s+0.0001\right)}$$

For experimentation, it is desired to build an automatic dispensing system to make the pain level constant as shown in figure below. It follows that ideally the injection system transfer function has to be

$$M(s)=\frac{1}{G(s)}$$

to obtain an overall transfer function M(s)G(s) $\approx$ 1. However, for implementation purposes M(s) must have at least one more pole than zeros. Find a suitable transfer function, M(s) by inverting G(s) and adding poles that are far from the imaginary axis.

$$\scriptscriptstyle \begin{matrix} && \text{Infusion Pump} && \text{Human response}\\ \text{Constant} &&&&&& \text{Constant}\\ \text{Infusion} & \mathrm{\rightarrow} & \text{M(s)} & \mathrm{\rightarrow} & \text{G(s)} & \mathrm{\rightarrow} & \text{back}\\ \text{Rate} &&&&&& \text{pain}\\ \end{matrix}$$

Find the area of the following region. The region inside the limaçon $r=4-2 \cos \theta$

Find the limit of the following sequence or determine that the sequence diverges. $\left\{\frac{(-1)^{n}}{n}\right\}$

For the following exercise, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. Use the method of shells to find the volume of an ellipse $\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1$ rotated around the x-axis.

Find the polar equation of the tine tangent to the polar curve $r=\cos \theta+\sin \theta$ at the origin, and then find the slope of this tangent line.

The specific rotation of l-dopa in water (at $15^{\circ} \mathrm{C}$) is −39.5. A chemist prepared a mixture of l-dopa and its enantiomer, and this mixture had a specific rotation of −37. Calculate the % ee of this mixture.

State the slope and the y-intercept of the graph of each equation. $y=x+9$

For each of the transfer functions shown below, find the locations of the poles and zeros, plot them on the s-plane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on).

$$\text { a. } T(s)=\frac{2}{s+2}$$

$$\text { b. } T(s)=\frac{5}{(s+3)(s+6)}$$

$$\text { c. } T(s)=\frac{10(s+7)}{(s+10)(s+20)}$$

$$\text { d. } T(s)=\frac{20}{s^{2}+6 s+144}$$

$$\text { e. } T(s)=\frac{s+2}{s^{2}+9}$$

$$\text { f. } T(s)=\frac{(s+5)}{(s+10)^{2}}$$

Find the limit of the following sequence or determine that the sequence diverges. $\left\{\int_{1}^{n} x^{-2} d x\right\}$

A uniform string of length 2.5 m and mass 0.01 kg is placed under a tension 10 N. (a) What is the frequency of its fundamental mode? (b) If the string is plucked transversely and is then touched at a point 0.5 m from one end. what frequencies persist?

List the minor matrix $M_{ij}$, and calculate the cofactor $A_{i j}=(-1)^{i+j} \operatorname{det}\left(M_{i j}\right)$ for the matrix A given by

$$A=\left[\begin{array}{rrrr} 2 & -1 & 3 & 1 \\ 4 & 1 & 3 & -1 \\ 6 & 2 & 4 & 1 \\ 2 & 2 & 0 & -2 \end{array}\right]$$

$M_{11}$

(a) You may have observed that water waves advancing toward shore have their wavefronts almost always parallel to the shoreline independent of the direction of the wind? Noting the fact that the velocity of waves in water decreases as the depth of the water decreases, use Huygens' Principle to explain this phenomenon. (b) To make the analysis of (a) more specific, assume that waves, initially traveling in the x direction, enter a region in which their speed v has a systematic variation with the distance y perpendicular to the direction of travel. (For example, x could be the direction parallel to the shore, and y would then be the direction perpendicular to the shore.) Show that the direction of the waves will begin to follow the arc of a circle of radius R, such that $R=\frac{v}{d v / d y}$

Find the trigonometric Fourier series for a periodic function f(t) that is equal to $t^{2}$ over the period from t=0 to t=2.

Combing your hair leads to excess electrons on the comb. How fast would you have to move the comb up and down to produce red light?