Question 1) Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) E R if and only if a) a is taller than b. b) a and b were born on the same day. c) a has the same first name as b. d) a and b have a common grandparent.

(1 point) The slope field for a population P modeled by dP/dt = 2.5P – 5P2 is shown in the figure below. (a) On a print-out of the slope field, sketch three non-zero solution curves showing different types of behavior for the population P. Give an initial condition that will produce each: P(0) = P(0) = P(0) = and (b) Is there a stable value of the population? If so, give the value; if not, enter none: Stable value = (c) Considering the shape of solutions for the population, give any intervals for which the following are true. If no such interval exists, enter none, and if there are multiple intervals, give them as a list. (Thus, if solutions are increasing when P is between 1 and 3, enter (1,3) for that answer; if they are decreasing when P is between 1 and 2 or between 3 and 4, enter (1,2).(3,4). Note that your answers may reflect the fact that P is a population.) P is increasing when P is in Pis decreasing when P is in Think about what these conditions mean for the population, and be sure that you are able to explain that. In the long-run, what is the most likely outcome for the population? P→ (Enter infinity if the population grows without bound.) Are there any inflection points in the solutions for the population? If so, give them as a comma-separated list (e.g., 1,3); if not, enter none. Inflection points are at P = Be sure you can explain what the meaning of the inflection points is for the population. (d) Sketch a graph of dP/dt against P. Use your graph to answer the following questions. When is dP/dt positive? When P is in When is dP/dt negative? When P is in (Give your answers as intervals or a list of intervals.) When is dP/dt zero? When P = (If there is more than one answer, give a list of answers, e.g., 1,2.) (Give your answers as intervals or a list of intervals.) When is dP/dt zero? When P = (If there is more than one answer, give a list of answers, e.g., 1,2.) When is dP/dt at a maximum? When P = Be sure that you can see how the shape of your graph of dP/dt explains the shape of solution curves to the differential equation.

(7.10 For a wave characterized by the electric field E(z.t) = ħax cos(wt – kz) + ġ ay cos(wt – kz + 8) identify the polarization state, determine the polarization angles (y,x), and sketch the locus of E(0,t) for each of the following cases: (a) dx = 3 V/m, ay 3 V/m, ay = 4 V/m, and 8 = 0 (b) az = 3 V/m, ay = 4 V/m, and 8 = 180° (c) az = 3 V/m, ay = 3 V/m, and 8 = 45° (d) ax = 3 V/m, ay = 4 V/m, and 8 = -135°

Marvin’s Magical Potions and More shop is located in a small rented store front. Each month Marvin pays $1,200 in rent, around $300 in utilities, another $100 in taxes and fees. His one salesperson earns roughly $1,000 per month independent of how much she’s there. His potions run approximately 50 cents per unit to produce. (The cost of the spell that give them their value is unfortunately impossible to quantify.) Further, each customer that enters is offered a free Tarot reading. A) Articulate the marginal costs associated with selling a single potion. B) Justify why you excluded the remaining costs. C) Assuming he can sell his potions for approximately $10 per potion, should he? Explain.

6.18 The file Property Taxes contains the property taxes per capita for the 50 states and the District of Columbia. Decide whether the data appear to be approximately normally distributed by a. comparing data characteristics to theoretical properties. b. constructing a normal probability plot.

If the current flowing through a 75 mH inductor has the waveform shown in Fig. 7.46, (a) sketch the voltage which develops across the inductor terminals for t ≥0, assuming the passive sign convention; and (b) calculate the voltage at t =1s ,2.9s , and 3.1 s.                                

The lumen output was determined for each of I = 3 different brands of lightbulbs having the same wattage, with ) = 8 bulbs of each brand tested. The sums of squares were computed as SSE = 4774.2 and SST = 594.9. You can use the Distribution Calculators page in SALT to find critical values and/or p-values to answer parts of this question. State the hypotheses of interest (including word definitions of parameters). u; = sample average lumen output for brand i bulbs Holly = M2 = 43 Ha: all three u 's are unequal Ομ, = true average lumen output for brand i bulbs Ho: My & M₂ = M3 Ha: at least two ;'s are equal O Hj = true average lumen output for brand i bulbs Ho: M1 = M2 = 43 Ha: at least two u;'s are unequal O u; = sample average lumen output for brand i bulbs Ho: My & M₂ 7 M3 Ha: all three u;'s are equal Use the Ftest of ANOVA (a = 0.05) to decide whether there are any differences in true average lumen outputs among the three brands for this type of bulb by obtaining as much information as possible about the P-value. (Round your answer to two decimal places.) f=