Provide a brief explanation of Plato's doctrine of Essences, Ideas, or Forms.

Ultimate reality, for Plato, is spiritual. The realm of spirits which Plato calls The One is where ideal forms or absolutes exist. The ideal forms give rise to our physical world because our physical world is nothing more than a shadow or imitation of the complete forms found in the spiritual realm. In the physical realm, we can recognize a chair as being a chair, only because the ideal chair exists in the spiritual realm and pre-existed the physical world's chair. Without the existence of the ideal chair the material chair in our world, which is just a shadow according to Plato, could not exist. It is as if we exist in a cave and can only see shadows of objects beyond the cave. We would accept the shadows as real, not realizing the shadow is depended on the object beyond the cave.

Find the derivative of the following function: f(x)= (3-5x-x^2)/(x^2-6)

To solve for the derivative of f(x) you need to use quotient rule. I remember quotient rule as the following ((High D *Low) - (Low D *High))/(Low D)^2: Using Quotient Rule the derivative is: f'(x) = (5x^2+6x+30)/(x^2-6)^2}

Find the Solution to the system of equations below: 2x+3y+1=0 -2x+6y+2=0

There are multiple ways to solve this problem which include graphing, substitution, elimination and so on. The best way based on the problem above is to use elimination then substituting the answer for y back into one the original equations and then checking to make sure that both values for x and y make both equations above true. steps: 1) Add the two equations thus eliminating the x variable. 9y+3=0 2) solve for y. y =-1/3 3)Plug y back into one of the equations above and solve for x. 2x+3(-1/3)+1 =0 x=0 4) Check both x and y values in the orignal equations to make sure the statement is true. 2(0)+3(-1/3)+1=0 true -2(0)+6(-1/3)+2=0 true