solve the given problems Let f(x) = ln(1 + x). i. Calculate the first, second, third, and

solve the given problems

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Let f(x) = ln(1 + x). i. Calculate the first, second, third, and fourth derivatives of f. ii. Using these derivatives, obtain the Taylor polynomials of degree n of f at b = 0, for n equals to zero, one, two, and three; i.e., Po,b(x), P₁,b(x), P2,6(x), P3,6(x), with b 0. = iii. Use the direct subtraction Rnb (x) = ln(1 + x) − Pn,b(x), to evaluate the errors when approximating f(0.5) using Pn,b (0.5) where b 0, 1, 2, 3. = 0 and n = |Rn+1,b(x)| Rn,b(x)| for n = - = 0, 1, 2, where b 0 and x 0.5 using the results you get from point iii. 2. iv. Calculate v. Write the residual formula Rn,b(x) using only the n + 1 derivative of f for n 2, 3. - vi. Using the formula you get above, give an estimate of |Rn,b(0.25)| Rnb (0.5)| where 0 and n is just an estimate, you should not do this task by using direct subtraction as in iii. = 2 or n = 3. As this Let f(x) = ln(1 + x). i. Calculate the first, second, third, and fourth derivatives of f. ii. Using these derivatives, obtain the Taylor polynomials of degree n of f at b = 0, for n equals to zero, one, two, and three; i.e., Po,b(x), P₁,b(x), P2,6(x), P3,6(x), with b 0. = iii. Use the direct subtraction Rnb (x) = ln(1 + x) − Pn,b(x), to evaluate the errors when approximating f(0.5) using Pn,b (0.5) where b 0, 1, 2, 3. = 0 and n = |Rn+1,b(x)| Rn,b(x)| for n = - = 0, 1, 2, where b 0 and x 0.5 using the results you get from point iii. 2. iv. Calculate v. Write the residual formula Rn,b(x) using only the n + 1 derivative of f for n 2, 3. - vi. Using the formula you get above, give an estimate of |Rn,b(0.25)| Rnb (0.5)| where 0 and n is just an estimate, you should not do this task by using direct subtraction as in iii. = 2 or n = 3. As this

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i To calculate the derivatives of fx ln1x we will use the chain rule and the derivative of the natural logarithm function First derivative fx ddx ln1x ... View the full answer