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P05 - Project 5 MAC 2311 1 In an environment where the...

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Project 5 MAC 2311 1. In an environment where the temperature is held constant, the relationship between the pres- sure ρ and volume ν of a gas is given by the function ν ( ρ ) = c ρ on ( 0 , ) , where c is a constant; suppose the units are adjusted so that c = 1 . Give a quick sketch of the function ν ( ρ ) on the axes below, and label the points on the graph that correspond to ρ = 1 2 , ρ = 1 , ρ = 3 2 , and ρ = 2 . Call these points A , B , C , and D respectively. Draw the lines between points A and D , between points B and D , and between points C and D . Then calculate the slope of each of the three lines. Are the above slopes enough information to deduce the exact slope of the tangent line at point D ? Make a guess below. Let’s calculate the slope of the tangent line at point D using limits: If you take point D and another generic point ( ρ, ν ( ρ )) on the curve, then what is the slope m ( ρ ) of the line between them in terms of ρ ? Simplify your formula. What value does the slope approach as you choose the generic point closer and closer to D (by making ρ closer and closer to 2 ) ? Write the equation of the line tangent to the graph of
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