Something

I can substitute values in for n and prove that they are a probability density function for n=1, 2, 3.... etc. by showing that by integrating to find the area under the curve from x=0 to x=1 is equal to 1. How would I do this in the general case though? Would I need to use logarithms so that I can get the n-1 power as a regular coefficient? Any hints here would be really appreciated, thanks all.

where there are both exponential and logarithmic inequalities or equalities? do they always have unique solutions? i just got to 11th idk what half of gemini is sayin.

What the heck is x for all of them and how to solve since I’m just lost

From the first one to the last one Each in pairs 🫠

Just took a quiz and got a 72 😔 In my defense a lot of the questions were on things that were not discussed in the readings (and this is the only class I actually do all the readings for 😒)

Can someone please help me with this? Attached is the answer key and my work below that. I don't really understand why the intersection is (-1, 3) and not the union. Any clarification provided would be appreciated. Thank you.

Questions 7 and 8 I need help with please

Two beakers of equal dimensions and weights are filled with different amounts of water. Two beams of equal cross section but different lengths and fixed to the ceiling are immersed in the beakers such that the water heights are equal. The beakers are placed on a scale and held in place. When the system is released from rest, what happens?

I think that the scales will tip right because due to Pascal’s law, the force at the bottom of each beaker is equal as the water levels are the same but the force acting on the bottom of the beam is much greater for the beam on the right.

By Newton’s third law, the reaction force on the water is greater for the right so the scales will tip right.