(a) Find the linear approximation of the function f(x) = /1 – x about a = 0. Use it to approximate the square rootof 9/10.(b) You are eating a cylindrical stack of pancakes that has a radius and height of 5 cm. You want to cover your wholestack in a thin layer of syrup that is 1/10 cm thick. Use differentials to estimate the change in volume of your stack ofpancakes after adding the layer of syrup. Use that the volume of a cylinder is V = Tr2h.

Answered: (a) Find the linear approximation of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(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.
(b) You are eating a cylindrical stack of pancakes that has a radius and height of 5 cm. You want to cover your whole
stack in a thin layer of syrup that is 1/10 cm thick. Use differentials to estimate the change in volume of your stack of
pancakes after adding the layer of syrup. Use that the volume of a cylinder is V = Tr2h.

Expert Solution

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Step 2

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

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

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

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

Now,

$f (x) = \sqrt{1 - x} \Rightarrow f' (x) = \frac{d}{d x} (\sqrt{1 - x}) \Rightarrow f' (x) = \frac{1}{2 \sqrt{1 - x}} \frac{d}{d x} (1 - x) [\sin c e \frac{d}{d x} (\sqrt{g (x)}) = \frac{1}{2 \sqrt{g (x)}} \times g' (x)] \Rightarrow f' (x) = \frac{- 1}{2 \sqrt{1 - x}}$

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

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

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

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