Explain the geometric interpretation of exact differential equations. talk about gradients, the multivariable chain rule, parametric curves and velocity or tangent vectors. what do these have to do with the condition for exactness of a differential equation? use a specific example and draw pictures to elaborate.

The radical equation 2+√2x-3 = √x+7 has a solution set [x= a0} and an extraneous root x = a1.

In a singing competition, there are 150 participants. at the end of each round, 40% of the participants are eliminated. how many participants are left after n rounds?

The architect's side view drawing of a saltbox-style house shows a post that supports the roof ridge. the support post is 8 ft tall. the distance from the front of the house to the support post is less than the distance from the post to the back of the house. how far from the front of the house is the support post positioned?