How would you define a point with polar coordinates (3, pi/6) in cartesian coordinates?
We are converting polar coordinates to cartesian coordinates here, so we must use the conversion equations as such: x=rcos(theta), y=rsin(theta). Plugging our values for r=3 and theta=pi/6 in accordingly, we find that x=(3)(sqrt(3)/2) and y=3(1/2).
Describe, in physical terms, the meaning of a derivative, and why we need them when studying natural phenomena.
A derivative is the rate of change of a specific variable or parameter over another unit, typically represented as "x" or as time. Derivatives are the way in which we model the change in a variable that is also applicable to curves, whereas in the case of lines, the "slope" is all we would need. As such, every derivative is an approximation; none is technically numerically "perfect," and in engineering, the degree to which our approximations are precise tends to determine the durability of a system. Almost no natural phenomenon, be it weather, electricity, water flow, economics, subatomic physics, or otherwise, can be modeled linearly - there are simply too many variables affecting any given thing. As such, we need calculus in order to perform the kinds of math that we use in the real world.
What was the significance of the year 1453, both in the context of Europe as a whole as well as in shaping the history of the Eastern European region?
1453 is the year in which the city of Constantinople, or Byzantium, fell to the Ottoman Turks, marking the end of the Eastern Roman Empire and thus the final end of the Roman Empire. The Turks control the city (as Istanbul) to this day, and as such the city still represents the de facto dividing line between what we consider Europe and Asia. As a result, notions of what defined the "European identity" gravitated further towards exclusively Western ideals, and the presence of Islamic culture and identity found its way into the cultures of Greece and many other Balkan nations in the region.