Let's solve the problem step by step, **explaining each step** and each **option** for the error identification: --- ## **Step 1: Find the Derivative** Given \( f(x) = x^{1/4} \): - The derivative using the power rule is: \[ f'(x) = \frac{1}{4} x^{-3/4} \] - This matches the derivative used in the work shown. --- ## **Step 2: Find \( f'(a) \) where \( a = 81 \)** - Substitute \( a = 81 \): \[ f'(81) = \frac{1}{4} \times (81)^{-3/4} \] - Let's compute \( (81)^{-3/4} \): - \( 81^{1/4} = 3 \) because \( 3^4 = 81 \). - \( 81^{-3/4} = (81^{1/4})^{-3} = 3^{-3} = \frac{1}{27} \). So, \[ f'(81) = \frac{1}{4} \times \frac{1}{27} = \frac{1}{108} \] - The derivative calculation is correct in the work shown. --- ## **Step 3: Find \( \Delta x \)** - \( a = 81, \quad b = 81.4 \) - Change in \( x \) is: \[ \Delta x = b - a = 81.4 - 81 = 0.4 \] **NOTE:** In the work shown, they used \(-0.4\) instead of \(+0.4\). This is a sign error. --- ## **Step 4: Linear Approximation Formula** \[ \Delta f \approx f'(a) \Delta x \] Plug values in: \[ \Delta f \approx \frac{1}{108} \times 0.4 \] \[ \Delta f \approx 0.00370 \] - **But in the work shown, they used \(-0.4\)**, so their value is negative (which is not correct). --- ### **Identifying the First Error** Let's analyze each option: 1. **The value of \( f'(a) \) should be \( f'(a) = 6.75 \).** - \( f'(a) = \frac{1}{108} \), not 6.75. So this is incorrect. 2. **The change in \( x \) should be \( \Delta x = 0.4 \).** - **This is correct.** They incorrectly used \(-0.4\). 3. **The value of \( f'(a) \) should be \( f'(a) = -6.75 \).** - This is not true. 4. **The derivative of \( f \) should be \( x^{-3/4} \).** - The derivative is \( \frac{1}{4} x^{-3/4} \), not just \( x^{-3/4} \). 5. **The derivative of \( f \) should be \( \frac{1}{4}x^{3/4} \).** - This is incorrect. The derivative is \( \frac{1}{4} x^{-3/4} \). ### **Correct Option:** - **The change in \( x \) should be \( \Delta x = 0.4 \).** --- ## **Step 5: Compute the Linear Approximation** \[ \Delta f \approx \frac{1}{108} \times 0.4 \] \[ \Delta f \approx 0.00370 \] (to five decimal places: \( 0.00370 \)) --- ## **Step 6: Compute Actual Change and Error** ### **Actual change:** \[ \Delta f_\text{actual} = f(81.4) - f(81) \] Calculate: - \( f(81.4) = (81.4)^{1/4} \) - \( f(81) = (81)^{1/4} = 3 \) Calculate \( (81.4)^{1/4} \): - Using a calculator: \( 81.4^{0.25} \approx 3.00123 \) So, \[ \Delta f_\text{actual} \approx 3.00123 - 3 = 0.00123 \] ### **Compute the Error:** \[ \text{error} = |\Delta f_\text{actual} - \Delta f_\text{approx}| \] \[ \text{error} \approx |0.00123 - 0.00370| = 0.00247 \] --- ## **SUMMARY OF ANSWERS** ### **First box:** \[ \Delta f \approx 0.00370 \] ### **Second box (error):** \[ \text{error} \approx 0.00247 \] --- ## **EXTRA: Explanation of Each Option** 1. **\( f'(a) = 6.75 \):** Incorrect, derivative is \( \frac{1}{108} \). 2. **\( \Delta x = 0.4 \):** **Correct.** Error was using \(-0.4\) instead. 3. **\( f'(a) = -6.75 \):** Incorrect. 4. **Derivative is \( x^{-3/4} \):** Not correct, missing \( \frac{1}{4} \) factor. 5. **Derivative is \( \frac{1}{4} x^{3/4} \):** Incorrect exponent sign. --- ## **Final Answers to Fill In** **Linear Approximation:** \[ \Delta f \approx 0.00370 \] **Error:** \[ \text{error} \approx 0.00247 \] **Correct Error Option:** **The change in \( x \) should be \( \Delta x = 0.4 \).**

Let's carefully analyze and solve the problem step by step, providing clear explanations at each stage: --- ### **Understanding the problem** Given the function \( f(x) = x^{1/4} \), we want to estimate the change in \( f \) (denoted as \( \Delta f \)) when \( x \) changes from \( a = 81 \) to \( b = 81.4 \). The method used is **Linear Approximation**, which approximates the change in the function as: \[ \Delta f \approx f'(a) \times \Delta x \] where \( \Delta x = b - a \). --- ### **Step 1: Find the derivative \( f'(x) \)** Using the power rule: \[ f(x) = x^{1/4} \implies f'(x) = \frac{1}{4} x^{(1/4) - 1} = \frac{1}{4} x^{-3/4} \] This derivative tells us how the function changes with respect to \( x \). --- ### **Step 2: Calculate \( f'(a) \) at \( a = 81 \)** Substitute \( a = 81 \): \[ f'(81) = \frac{1}{4} \times 81^{-3/4} \] Now, compute \( 81^{-3/4} \): - First, find \( 81^{1/4} \): Since \( 3^4 = 81 \), then \( 81^{1/4} = 3 \). - Then, \( 81^{-3/4} = (81^{1/4})^{-3} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27} \). So, \[ f'(81) = \frac{1}{4} \times \frac{1}{27} = \frac{1}{108} \] which is approximately \( 0.009259 \). --- ### **Step 3: Determine \( \Delta x \)** Given \( a = 81 \) and \( b = 81.4 \): \[ \Delta x = 81.4 - 81 = 0.4 \] --- ### **Step 4: Estimate \( \Delta f \) using Linear Approximation** Calculate: \[ \Delta f \approx f'(81) \times \Delta x = \frac{1}{108} \times 0.4 \] Perform the multiplication: \[ \frac{1}{108} \times 0.4 = \frac{0.4}{108} \approx 0.0037037 \] Rounded to five decimal places: \[ \boxed{\Delta f \approx 0.00370} \] --- ### **Step 5: Find the actual change \( f(81.4) - f(81) \) and compute the error** Calculate \( f(81.4) \): \[ f(81.4) = (81.4)^{1/4} \] Using a calculator: \[ 81.4^{1/4} \approx 3.00123 \] Since \( f(81) = 3 \), the actual change is: \[ \Delta f_\text{actual} \approx 3.00123 - 3 = 0.00123 \] **Error in approximation:** \[ \text{Error} = |\text{Actual} - \text{Approximate}| = |0.00123 - 0.00370| \approx 0.00247 \] --- ### **Summary of the key points:** - The **estimated change** in \( f \): \[ \boxed{\Delta f \approx 0.00370} \] - The **error** in the linear approximation: \[ \boxed{\approx 0.00247} \] --- ### **Identifying the first error in the original estimate:** From the options: - **Option 1:** *The value of \( f'(a) \) should be \( 6.75 \).* — **Incorrect**, because \( f'(81) = 1/108 \approx 0.009259 \). - **Option 2:** *The change in \( x \) should be \( 0.4 \).* — **Correct**, because \( 81.4 - 81 = 0.4 \). - **Option 3:** *The value of \( f'(a) \) should be \( -6.75 \).* — Incorrect. - **Option 4:** *The derivative of \( f \) should be \( x^{-3/4} \).* — The derivative is \( \frac{1}{4} x^{-3/4} \), so this is incomplete. - **Option 5:** *The derivative of \( f \) should be \( \frac{1}{4} x^{3/4} \).* — Incorrect, since the derivative involves a negative exponent. **Therefore, the first error is that the change in \( x \) was incorrectly taken as \(-0.4\) instead of \( +0.4 \).** --- ### **Final notes:** - **Estimated change \( \Delta f \):** approximately **0.00370**. - **Error in the approximation:** approximately **0.00247**. - **First mistake in the original estimate:** using the wrong sign for \( \Delta x \), i.e., using \(-0.4\) instead of \( +0.4 \). --- **In words:** The linear approximation estimates the change in \( f \) as roughly 0.00370 when \( x \) increases from 81 to 81.4. The actual change is about 0.00123, so the approximation slightly overestimates the true change. The main mistake in the original estimate was using a negative change in \( x \) (−0.4) instead of the correct positive change (0.4), which affects the sign and accuracy of the estimate.