(a) Find the linear approximation of the function f(x) = /1– x about a = 0. Use it to approximate the square root of 9/10.

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(a) The given function is $f\left(x\right)=\sqrt{1-x}$.

We have to find the linear approximation of $f\left(x\right)=\sqrt{1-x}$ about $a=0$.

We know that the linear approximation of a function $f\left(x\right)$ about a is given by $f\left(x\right)\approx f\left(a\right)+f\text{'}\left(a\right)\left(x-a\right)$.

Here, we have $f\left(x\right)=\sqrt{1-x}$ and $a=0$.

Now,

$$\begin{gathered} f\left(x\right)=\sqrt{1-x} \\ \Rightarrow f\text{'}\left(x\right)=\frac{d}{dx}\left(\sqrt{1-x}\right) \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{2\sqrt{1-x}}\frac{d}{dx}\left(1-x\right) \left[\sin ce \frac{d}{dx}\left(\sqrt{g\left(x\right)}\right)=\frac{1}{2\sqrt{g\left(x\right)}}\times g\text{'}\left(x\right)\right] \\ \Rightarrow f\text{'}\left(x\right)=\frac{-1}{2\sqrt{1-x}} \end{gathered}$$

Therefore, when $a=0$, by the linear approximation formula

$\begin{array}{rcl}f\left(x\right)& \approx & f\left(0\right)+f\text{'}\left(0\right)\left(x-0\right)\\ & =& \sqrt{1-0}+\frac{-1}{2\sqrt{1-0}}x\\ & =& 1-\frac{x}{2}\end{array}$

Thus, the linear approximation of the function $f\left(x\right)=\sqrt{1-x}$ about $a=0$ is $\begin{array}{rcl}f\left(x\right)& \approx & 1-\frac{x}{2}\end{array}$.