IGNOU MTE 10 2018 - Use the polynomial in part (i) to approximate .

Question

Use the polynomial in part (i) to approximate
. $\int_{0}^{0.1} (1+x)^\frac{1}{2} dx.$

14 Feb 2018

Answer :

Word Count : 291

To approximate the integral $\int_0^{0.1} (1 + x)^{1/2} \, dx$ using a polynomial, we can use a Taylor series expansion of the integrand at $x = 0$. This will allow us to approximate $(1 + x)^{1/2}$ as a polynomial. ### Step 1: Taylor Series Expansion of $(1 + x)^{1/2}$ The general Taylor series for $(1 + x)^n$ around $x = 0$ is: \[ (1 + x)^n = 1 + n x + \frac{n(n-1)}{2} x^2 + \frac{n(n-1)(n-2)}{6} x^3 + \cdots \] For $n = \frac{1}{2}$, we get the following expansion for $(1 + x)^{1/2}$: \[ (1 + x)^{1/2} = 1

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