Prove rigid motions are:

a. continuous

b. one-to-one

c. closed under composition

What is saving in open economy? Discuss what are differences of saving in closed and open economy?

1. Theoretically prove the boundary conditions for waves in closed-open tubes. Please provide pictures.

What are the four processes that define a closed-loop BPM cycle? Give example for each.

Consider the function ?(?,?)=3?^2−4?+??^2 on the closed region ?={(?,?):−1≤?≤1 and −1≤?≤1}R={(x,y):−1≤x≤1 and −1≤y≤1}.

(a) Find all critical points of ?(?,?)f(x,y) in the region ?R, if any, and classify them (local maximum, local minimum, or saddle point).

(b) Determine the absolute maximum and absolute minimum of ?(?,?)f(x,y) on the closed region ?R, and all points at which they occur.

1. Describe the conditions under which an op-amp circuit will oscillate. Provide a technique to avoid undesirable oscillation.

2. Explain why the gain-bandwidth product (GBW) is limited for any amplifier, and why GBW is constant for a fully compensated voltage feedback op-amp. If a fully compensated op-amp has GBW = 4 MHz, and its closed-loop amplifier circuit has gain (closed-loop gain) = 100, what is the loop gain at 100 Hz?

The figure shows a 11.8 V battery and four uncharged capacitors of capacitances C1 = 1.16 μF,C2 = 2.31 μF,C3 = 3.26 μF, and C4 = 4.19 μF. If only switch S1 is closed, what is the charge on (a) capacitor 1, (b) capacitor 2, (c) capacitor 3, and (d) capacitor 4? If both switches are closed, what is the charge on (e) capacitor 1, (f) capacitor 2, (g) capacitor 3, and (h) capacitor 4?

For each of the scenarios below, determine how the total mass of the described object or system is changing, and say whether the scenario Increases the Mass, Decreases the Mass, or that the Mass Remains Unchanged:

A hot stone cooling down

A closed, thermally isolated box containing a controlled exothermic nuclearreaction

A closed, thermally isolated box containing a controlled exothermic chemicalreaction

A glass full of ice cubes melting

A compressed spring slowly relaxing

A car batter being charged

Suppose that k is a field which is not algebraically closed. a. Show that if I ⊂ k[x1, . . . , xn ] is maximal, then V(I) is either empty or a point in kn . Hint: Examine the proof of Theorem 11. b. Show that there exists a maximal ideal I in k[x1, . . . , xn ] for which V(I) = ∅. Hint: See the previous exercise. c. Conclude that if k is not algebraically closed, there is always a maximal ideal of k[x1, . . . , xn ] which is not of the form <x1 − a1, . . . , xn − an >