### Transcription for Educational Purposes**Task: Estimation Using Differentials****b.** Use the differential you've just calculated to estimate the value of \(\sqrt{9.2}\).### ExplanationThis task involves using differentials—a concept from calculus—to estimate the square root of 9.2. Differentials provide an approximation method based on linear approximations of functions. Here's a step-by-step approach typically used:1. **Identify the Function:** Here the function of interest is \(f(x) = \sqrt{x}\).2. **Find the Derivative:** The derivative, \(f'(x)\), of \(f(x) = \sqrt{x}\) is \(f'(x) = \frac{1}{2\sqrt{x}}\).3. **Select a Point \(a\):** Choose a point \(a\) close to 9.2 where the square root is easy to calculate, typically \(a = 9\).4. **Calculate \(f(a)\) and \(f'(a)\):** \[ f(9) = \sqrt{9} = 3 \] \[ f'(9) = \frac{1}{2 \times 3} = \frac{1}{6} \]5. **Determine the Change, \(dx\):** The difference between 9.2 and 9, which is \(dx = 9.2 - 9 = 0.2\).6. **Compute the Differential:** Use the formula \(dy = f'(a) \cdot dx\): \[ dy = \frac{1}{6} \times 0.2 = \frac{0.2}{6} \approx 0.0333 \]7. **Estimate the Value:** \[ \sqrt{9.2} \approx f(a) + dy = 3 + 0.0333 = 3.0333 \]This is a linear approximation and serves as an estimate for \(\sqrt{9.2}\). **Problem Statement:**1. Given \( y = \sqrt{x - 4} \). a. Evaluate \( dy \) when \( x = 13 \) and \( dx = 0.2 \).**Explanation:**To solve this problem, one would typically use differentiation to find \( dy \) in terms of \( dx \). The function \( y = \sqrt{x - 4} \) suggests finding the derivative to determine how changes in \( x \) affect \( y \).

Name: ### Transcription for Educational Purposes**Task: Estimation Using Di
Uploaded: 2024-03-20T06:46:42.000Z
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Description: ### Transcription for Educational Purposes**Task: Estimation Using Differentials****b.** Use the differential you've just calculated to estimate the value of \(\sqrt{9.2}\).### ExplanationThis task involves using differentials—a concept from calculu