Let's verify the linear approximation \((1+2x)^{-4} \approx 1 - 8x\) at \(a=0\). --- ### **Step 1: Find the function and its derivative** Let \( f(x) = (1+2x)^{-4} \). #### **First, find \( f(a) \) at \( a=0 \):** \[ f(0) = (1+2 \cdot 0)^{-4} = 1^{-4} = 1 \] #### **Now, find \( f'(x) \):** Use the chain rule: \[ f(x) = (1+2x)^{-4} \] Let \( u = 1+2x \), so \( f(x) = u^{-4} \). \[ \frac{d}{dx}(u^{-4}) = -4u^{-5} \cdot \frac{d}{dx}(u) = -4(1+2x)^{-5} \cdot 2 = -8(1+2x)^{-5} \] So, \[ f'(x) = -8(1+2x)^{-5} \] #### **Evaluate \( f'(a) \) at \( a=0 \):** \[ f'(0) = -8(1+2\cdot0)^{-5} = -8 \cdot 1^{-5} = -8 \] --- ### **Step 2: Write the linear approximation** The linear approximation at \( x=a=0 \) is: \[ L(x) = f(0) + f'(0)(x-0) \] \[ L(x) = 1 + (-8)x = 1 - 8x \] --- ### **Step 3: State the interval** The linear approximation works best for \( x \) values near \( 0 \): \[ x \in \mathbb{R} \quad \text{(but most accurate for \( x \) near \( 0 \))} \] --- ## **Final Answer** \[ (1+2x)^{-4} \approx 1 - 8x, \quad \text{for } x \in \mathbb{R} \text{ (best for } x \approx 0\text{)} \] Or, you may write: \[ x \in (-\epsilon, \epsilon) \text{ for small } \epsilon \] --- **Summary:** The given linear approximation is verified. **Boxed answer:** \[ \boxed{x \in \mathbb{R} \text{ (best for values near } 0\text{)}} \]

Let's carefully verify the linear approximation of the function \(f(x) = (1+2x)^{-4}\) at \(a=0\), step-by-step, with detailed calculations. --- ### **Step 1: Define the function** Given: \[ f(x) = (1+2x)^{-4} \] --- ### **Step 2: Calculate \(f(a)\) at \(a=0\)** \[ f(0) = (1 + 2 \times 0)^{-4} = (1)^{-4} = 1 \] --- ### **Step 3: Find the derivative \(f'(x)\)** Using the chain rule: \[ f(x) = (u)^{-4} \quad \text{where} \quad u = 1 + 2x \] Derivative of \(f(x)\) with respect to \(x\): \[ f'(x) = \frac{d}{dx} u^{-4} = -4 u^{-5} \cdot \frac{du}{dx} \] Since \(u = 1 + 2x\), \[ \frac{du}{dx} = 2 \] Thus: \[ f'(x) = -4 (1 + 2x)^{-5} \times 2 = -8 (1 + 2x)^{-5} \] --- ### **Step 4: Evaluate \(f'(a)\) at \(a=0\)** \[ f'(0) = -8 (1 + 2 \times 0)^{-5} = -8 \times 1^{-5} = -8 \times 1 = -8 \] --- ### **Step 5: Write the linear approximation** The linear approximation of \(f(x)\) at \(a=0\) is: \[ L(x) = f(0) + f'(0) \times (x - 0) = 1 + (-8) x = 1 - 8x \] --- ### **Final Result:** **The linear approximation of \((1+2x)^{-4}\) at \(a=0\) is:** \[ \boxed{ \quad (1+2x)^{-4} \approx 1 - 8x } \] **This approximation is most accurate for values of \(x\) near 0.** --- ### **Summary:** - **Function value at \(a=0\):** \(f(0) = 1\) - **Derivative at \(a=0\):** \(f'(0) = -8\) - **Linear approximation:** \(\boxed{1 - 8x}\) This completes the detailed verification of the linear approximation.