Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$. If the Mean Value Theorem cannot be applied, explain why not. $f(x)=x^{6}$, [-1, 1]

(a) Explain why the Intermediate Value Theorem gives no information about the zeros of the function $f(x)=\ln \left(x^{2}+2\right)$ on the interval [-2, 2]. (b) Use technology to determine whether f has a zero on the interval [-2,2].

Let f be a function for which $0 \leq f(x) \leq 1$ for all x in [0, 1]. If f is continuous on [0, 1], show that there exists at least one number c in [0, 1] such that f(c)=c.

Find the first- and second-order Taylor polynomials for the given function f at the given point a. $f(x, y, z)=1 /\left(x^{2}+y^{2}+z^{2}+1\right), \mathbf{a}=(0,0,0)$

A fungal spore a. contains an embryonic organism.b. germinates directly into an organism.c. is always windblown, because none are motile.d. is most often diploid.e. Both b and c are correct.

What factors does Paul Krugman identify that supported the expansion of international trade in the 1800s?

A flow of saturated liquid R-410a at 200 kPa in an evaporator is brought to a state of superheated vapor at 200 kPa, 20$^{\circ} \mathrm{C}$. Assuming the process is reversible, find the specific heat transfer and specific work.

A professional society, with a membership of 45,000, is designing a study to evaluate their membership’s satisfaction with the type of sessions presented at the society’s annual meeting. In each of the following descriptions of the method of selecting participants in the survey, identify the type of sampling method used (simple random sampling, stratified sampling, or cluster sampling). a. The society has an alphabetical listing of all its members. They assign a number to each name and then using a computer software program they generate 1,250 numbers between 1 and 45,000. They select these 1,250 members for the survey. b. The society is interested in regional differences in its membership’s opinion. Therefore, they divide the United States into nine regions with approximately 5,000 members per region. They then randomly select 450 members from each region for inclusion in the survey. c. The society is composed of doctors, nurses, and therapists, all working in hospitals. There are a total of 450 distinct hospitals. The society decides to conduct onsite in-person interviews, so they randomly select 20 hospitals and interview all members working at the selected hospital.

A fluid of specific gravity 0.90 flows at a Reynolds number of 1500 in a 0.3-m-diameter pipeline. The velocity 50 mm from the wall is 3 m/s. Calculate the flow rate and the velocity gradient at the wall.

A bivalent is a. a homologous chromosome.b. the paired homologous chromosomes.c. a duplicated chromosome composed of sister chromatids.d. the two daughter cells after meiosis I.e. the two centrioles in a centrosome.

A centrifugal water pump has an impeller with backward curved vanes and an inner diameter of 0.1 m, an outer diameter of 0.25 m, and a blade height of 4 cm. It operates at 1200 rpm. Water enters the impeller at the blade angle of 50$^{\circ}$ and leaves at the blade angle of 30$^{\circ}$. The volume flow rate is 0.18m$^{3}$/s. Determine the shaft torque and power. Determine the pressure rise when the fluid velocity leaving the pump diffuser is the same as that entering.

Do Problem 11.24 for R-507a, which has a 1:1 mass ratio of R-125 and R-143a. The refrigerant R143a has a molecular mass of 84.041 kg/kmoland Cp = 0.929 kJ/kg-K.Problem 11.24A new refrigerant, R-410a, is a mixture of R-32 and R-125 in a 1:1 mass ratio. What are the overall molecular weight, the gas constant, and the ratio of specific heats for such a mixture?

For a given volume flow rate and piping system, will the pressure loss be greater for hot water or cold water? Explain.

The geometry of a centrifugal water pump is r$_{1}$ = 10 cm, r$_{2}$ = 20 cm, b$_{1}$ = b$_{2}$ = 4 cm, $\beta_{1}$ = 30$^{\circ}$, $\beta_{2}$ = 15$^{\circ}$, and it runs at speed 1600 rpm. Estimate the discharge required for axial entry, the power generated in the water in watts, and the head produced.