Suppose that the population dynamics of a species obeys a modified version of the logistic differential equation having the following form: $\frac{d N}{d t}=r\left(1-\frac{N}{K}\right)^{2} N$ where $r \neq 0 \text { and } K>0$ Apply the local stability criterion to the equilibrium $\hat{N}=K.$ What do you think your answer means about the stability of this equilibrium?

Two protons approach each other with different speeds. (a) Will the magnitude of the total momentum of the two-proton system be (1) greater than the magnitude of the momentum of either proton, (2) equal to the difference between the magnitudes of momenta of the two protons, or (3) equal to the sum of the magnitudes of momenta of the two protons? Why? (b) If the speeds of the two protons are 340 m/s and 450 m/s, respectively, what is the total momentum of the two-proton system?

At a distance of 12.0 m from a point source, the intensity level is measured to be 70 dB. At what distance from the source will the intensity level be 40 dB?

A parallel RLC circuit has $R=100 \mathrm{k} \Omega,$ L = 100 mH, and a $C=10 \mu \mathrm{F}$. Determine the value of Q, the resonant frequency, and the bandwidth. If $R=200 \mathrm{k} \Omega,$ how does that affect the values of Q resonant frequency, and the bandwidth?

Analyze the impact a nonnative species might have on a native species in terms of population dynamics.

Why is the copious production of $\Lambda^{0}$ and K particles very difficult to reconcile with their slow decay, without the concept of strangeness? How does strangeness provide a reconciliation?

Rationalize each denominator. Assume that all radicands represent positive real numbers. $\frac{5}{\sqrt{10}}$

Determine whether the matrix is orthogonally diagonalizable.

$$\left[\begin{array}{rrr} {3} & {2} & {-3} \\ {-2} & {-1} & {2} \\ {-3} & {2} & {3} \end{array}\right]$$

A passenger walks forward along the aisle of a train at a speed of 1.3 m/sec as the train moves along a straight track at a constant speed of 30.2 m/sec with respect to the ground. What is the passenger's speed relative to the ground? To the accuracy cited, do classical and relativistic predictions differ?

Graph the first-octant portion of each plane. x=3

Predict the major organic product of each of the following reactions. (a) Benzoic anhydride + methanol $\underrightarrow{\mathrm{H}_{2} \mathrm{SO}_{4}}$ (b) Acetic anhydride + ammonia (2 mol) $\longrightarrow$ (c) Phthalic anhydride + $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NH}$ (2 mol) $\longrightarrow$ (d) Phthalic anhydride + sodium hydroxide (2 mol) $\longrightarrow$

Prove that the matrix below is orthogonal for any value of $\theta$.

$$\left[\begin{array}{ccc} {\cos \theta} & {-\sin \theta} & {0} \\ {\sin \theta} & {\cos \theta} & {0} \\ {0} & {0} & {1} \end{array}\right]$$

Write inequalities to describe the region. The region between the yz-plane and the vertical plane x = 5

What happens to the $\gamma$ rays that are emitted in stellar nuclear reactions of the proton-proton or carbon cycle?