Find all the antiderivatives of the following function. Check your work by taking derivatives. f(x) = 2 sin x + 1

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) 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 $R=\frac{v}{d v / d y}$

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

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

$M_{11}$

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.

Identify whether each of the following factors will affect the rate of a reaction: (a) $K_{\mathrm{eq}}$ (b) $\Delta G$ (e) $E_{\mathrm{a}}$ (c) Temperature (f) $\Delta S$

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?

Determine $\cos \theta$ where $\theta$ is the angle between u and v. $u = 2i - 3j + k, \ v = i - 2j + 3k$

Suppose you measure the terminal voltage of a 3.200-V lithium cell having an internal resistance of $5.00\ \Omega$ by placing a $1.00-\mathrm{k}\ \Omega$ voltmeter across its terminals. (a) What current flows? (b) Find the terminal voltage. (c) To see how close the measured terminal voltage is to the emf, calculate their ratio.

What would be the maximum cost of a CFL such that the total cost (investment plus operating) would be the same for both CFL and incandescent 60-W bulbs? Assume the cost of the incandescent bulb is 25 cents and that electricity costs 10 cents/kWh. Calculate the cost for 1000 hours, as in the cost effectiveness of CFL example.

(a) The velocity of sound in a gas is proportional to the square root of the absolute temperature T. Use this fact, and the result of the previous problem, to show that when a thermal gradient exists in the vertical direction (z) sound waves will be turned initially with a radius of curvature $R=\frac{2 T}{d T / d z}$ (b) On a still day, the temperature of the atmosphere is found to decrease more or less linearly with height. Sketch the paths of "rays" of sound emitted from a source suspended high in the atmosphere. Assuming that the velocity of sound at ground level is 1110 ft/sec, estimate the horizontal distance at which an airplane flying at 15,000 ft firs becomes audible to an observer on the ground, if the temperature decreases by $1^{\circ} \mathrm{C}$ per 500-ft increase in altitude.