1. Construct a second order Taylor polynomial, p2(x) f(x) = 1/(x + 1), x [] using...

  

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1. Construct a second order Taylor polynomial, p2(x) f(x) = 1/(x + 1), x [] using x = 0 and find its error, R2(x). Perform this calculation by hand. 2. Using the same function, construct Lagrange polynomial using nodes x = {0,1}. Do this calculation by hand. Plot your exact function and Lagrange polynomial using matplotlib. 3. Optimised f(x) =sin(x) with initial guess x = 0.5. Stop your calculation when tol/error = 105. Plot your exact function and the optimization point. For this question, use Python programming. 1. Construct a second order Taylor polynomial, p2(x) f(x) = 1/(x + 1), x [] using x = 0 and find its error, R2(x). Perform this calculation by hand. 2. Using the same function, construct Lagrange polynomial using nodes x = {0,1}. Do this calculation by hand. Plot your exact function and Lagrange polynomial using matplotlib. 3. Optimised f(x) =sin(x) with initial guess x = 0.5. Stop your calculation when tol/error = 105. Plot your exact function and the optimization point. For this question, use Python programming.