Verify the given linear approximation at *a* = 0. Then use a graphing calculator or computer to determine the values of *x* for which the linear approximation is accurate to within 0.1. (Round your answers to three decimal places. Enter your answer using interval notation.)

(1 + 4*x*)−3 ≈ 1 − 12*x*

*x* ∈ 

(−0.080,0.080)
 
 

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.

---

### 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!

Verify the given linear approximation at a = 0. Then use a graphing calculator or computer to determine the values of x for which the linear approximation is accurate to within 0.1. (Round your answers to three decimal places. Enter your answer using interval notation.) (1 + 4x)−3 ≈ 1 − 12x x ∈ (−0.080,0.080)

Verify the linear approximation (1 + 4x)^(-3) ≈ 1 − 12x at a = 0 and find the interval of x where the approximation is accurate to within 0.1. Step-by-step solution includes Taylor expansion, numerical methods, and error analysis.

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