Question

b) i) Calculate the third-degree Taylor polynomial abou x_0=0 for f(x)=(1+x)^{1/2}

ii) Use the polynomial in part (i) to approximate \sqrt{1.1} and find a bound for the error involved.

iii) Use the polynomial in part (i) to approximate \int _{0}^{0.1}(1+x)^{1/2}dx.

12 Mar 2024
Answer :
Word Count : 593
Let's break down the problem step by step: --- ### b) i) Calculate the third-degree Taylor polynomial about \( x_0 = 0 \) for \( f(x) = (1+x)^{1/2} \) The third-degree Taylor polynomial for \( f(x) \) around \( x_0 = 0 \) is given by: \[ T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 \] #### 1. Calculate the derivatives of \( f(x) \): \[ f(x) = (1 + x)^{1/2} \] First derivative: \[ f'(x) = \frac{1}{2}(1 + x)^{-1/2} \] Second derivative: \[ f''(x) = -\frac{1}{4}(1 + x)^{-3/2} \] Third derivative: \[ f^{(3)}(x) = \frac{3}{8}(1 + ______ ___ _________ ______ ____ _________ _____ _____ ______ ______ __________ __________.
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