Answered: (1 + x) -3= 1 – 3x

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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Question

Verify the given linear approximation at a=0 . Then
determine the values of x for which the linear approximation is
accurate to within 0.1.

Expert Solution

Step 1

The linear approximation: $L (x)$ for a function $f (x)$ at $x = a$ is defined by the formula:

$L (x) \approx f (a) + f' (a) (x - a)$

Step 2

Calculate $f' (x)$ for the function: $f (x) = {(1 + x)}^{- 3}$ as follows.

$\begin{array}{rcl} f (x) & = & {(1 + x)}^{- 3} \\ f' (x) & = & \frac{d}{d x} {(1 + x)}^{- 3} \\ = & - \frac{3}{{(1 + x)}^{4}} \end{array}$

Substitute $f' (x) = - \frac{3}{{(1 + x)}^{4}}, a = 0$ in the equation: $L (x) \approx f (a) + f' (a) (x - a)$ to obtain the linear approximation of the function: $f (x) = {(1 + x)}^{- 3}$ at $a = 0$ .

$\begin{array}{rcl} L (x) & \approx & f (a) + f' (a) (x - a) \\ \approx & {(1 + 0)}^{- 3} - \frac{3}{{(1 + 0)}^{4}} (x - 0) \\ \approx & 1 - 3 x \end{array}$

Hence, it is proved that ${(1 + x)}^{- 3} \approx (1 - 3 x)$ .

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