sol6 - Chapter 6 1 Section 6.1 1 Below is a graph of a...

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Chapter 6 1. Section 6.1 1. Below is a graph of a person’s annual earnings and annual money spent per year, over the course of their lifetime. Let I ( t ) denote annual earnings and S ( t ) denote annual money spent. Will this person need to find a financial advisor? dollars/year time in years I ( t ) S ( t ) Solution: The person’s savings at year t is precisely I ( t ) - S ( t ). Notice that the number of years S ( t ) exceeds I ( t ) far outweighs the number of years the person saves money. Therefore, the person is in debt. From a more mathematical perspective, if we let N ( T ) denote the person’s networth after T years, it can be calculated as N ( T ) = T 0 ( I ( t ) - S ( t )) dt As you can see from the graph, most of the time I ( t ) - S ( t ) < 0. 2. Below is the graph of a function f ( x ). Find the approximate locations of any inflection points and the locations of the absolute maximum and minimum for the antiderivative F ( x ). You may assume F (0) = 0. 1 2 3 4 5 0 1 2 - 1 - 2 f ( x ) 1 0 . 6 3 Solution: Notice that at about x = 2 and x = 3 . 5, the graph of f ( x ) flattens out. More specifically, at x = 2, f ( x ) goes from decreasing to increasing. Since F ( x ) = f ( x ), this is the same as saying that F ( x ) goes from decreasing to increasing. Thinking back to calculus 1, this means that the derivative of F ( x ), i.e. F ( x ), goes from being negative to positive. Therefore, F ( x

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