Consider a pair of molecules: They have an energy (u) that is a function of their...

 

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Consider a pair of molecules: They have an energy (u) that is a function of their distance (r). There are two possibilities for this function: a. u= 0.0073 +12 0.41 b. u 2.5 exp(-0.68r2) Here, u is in zepto-Joules [zJ], whilst r is in nano-meters [nm]. In the vicinity of r = 1.0, formulate a Taylor Series for each of these functions: Specifically, present the first-order and second-approximations in terms of r (as such, invoke an intermediate variable here). Then employing Python, prepare the appropriate graph for each of these (there will be two graphs in total). Each graph should have the exact function, together with its first-order and second-order approximations. Except for lines for the functions themselves, also have symbols that emphasize r = 0.6 and r = 0.8; use the options for dashed or dotted lines, for diamond or square symbols, etc. Also, make sure that you display a legend. Ultimately in each case, discuss which approximation you recommend for r = 0.6 and r = 0.8 (just based on their appearance on the graph). Consider a pair of molecules: They have an energy (u) that is a function of their distance (r). There are two possibilities for this function: a. u= 0.0073 +12 0.41 b. u 2.5 exp(-0.68r2) Here, u is in zepto-Joules [zJ], whilst r is in nano-meters [nm]. In the vicinity of r = 1.0, formulate a Taylor Series for each of these functions: Specifically, present the first-order and second-approximations in terms of r (as such, invoke an intermediate variable here). Then employing Python, prepare the appropriate graph for each of these (there will be two graphs in total). Each graph should have the exact function, together with its first-order and second-order approximations. Except for lines for the functions themselves, also have symbols that emphasize r = 0.6 and r = 0.8; use the options for dashed or dotted lines, for diamond or square symbols, etc. Also, make sure that you display a legend. Ultimately in each case, discuss which approximation you recommend for r = 0.6 and r = 0.8 (just based on their appearance on the graph).

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Answer To formulate the Taylor series approximations for each function lets consider the firstorder and secondorder expansions around r 1 Well introduce an intermediate variable x r r to simplify the ... View the full answer