To verify the given linear approximation and determine the interval where it is accurate to within 0.1, let's go step by step:

1. Verify the Linear Approximation at a = 0:

The given function is $f(x) = (1 + 4x)^{-3}$ with the linear approximation $f(x) \approx 1 - 12x$ near $x = 0$.

Step 1.1: Find $f'(x)$:

$f(x) = (1 + 4x)^{-3}$ Using the chain rule: $f'(x) = -3(1 + 4x)^{-4} \cdot 4 = -12(1 + 4x)^{-4}.$

Step 1.2: Find $f(0)$ and $f'(0)$:

At $x = 0$: $f(0) = (1 + 4(0))^{-3} = 1.$ $f'(0) = -12(1 + 4(0))^{-4} = -12.$

Thus, the linear approximation at $a = 0$ is: $f(x) \approx f(0) + f'(0)(x - 0) = 1 - 12x.$ This confirms that the given linear approximation $f(x) \approx 1 - 12x$ is correct.

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2. Determine the Interval Where the Approximation is Accurate to Within 0.1:

We need to find the values of $x$ for which: $|f(x) - (1 - 12x)| < 0.1.$

Step 2.1: Compute $f(x) - (1 - 12x)$:

$f(x) - (1 - 12x) = (1 + 4x)^{-3} - (1 - 12x).$

Step 2.2: Solve $|(1 + 4x)^{-3} - (1 - 12x)| < 0.1$:

This requires numerical methods or computational tools to solve.

Let me calculate the precise interval numerically.import numpy as np from scipy.optimize import root_scalar

Define the function and the linear approximation

def f(x): return (1 + 4*x)**-3

def linear_approx(x): return 1 - 12*x

Define the error function

def error(x): return abs(f(x) - linear_approx(x)) - 0.1

Find the roots of the error function

Interval for search

left_bound = root_scalar(error, bracket=[-0.2, 0], method='brentq').root right_bound = root_scalar(error, bracket=[0, 0.2], method='brentq').root

(left_bound, right_bound)(-0.029054554808090834, 0.03601679332550696)(-0.029054554808090834, 0.03601679332550696)successThe interval where the linear approximation $(1 + 4x)^{-3} \approx 1 - 12x$ is accurate to within 0.1 is approximately:

$x \in (-0.029, 0.036)$

Would you like further details or have any questions? Here are 5 related questions to delve deeper:

1. How is the linear approximation derived from the Taylor expansion?
2. What happens to the error as $x$ moves further from $a = 0$?
3. Can we approximate higher-order terms for better accuracy?
4. How does the chain rule apply to derivatives in this problem?
5. Would this method work for functions with sharper curves?

Tip: Always visualize the approximation alongside the function to identify error bounds effectively!