The side of a square floor tile is measured to be 16 inches, with a possible error of 1/32 inch. Use differentials to approximate the possible error and the relative error in computing the area of the square. Step 1 Recall that the formula for the area of a square is A = x We are given that the side of the square floor tile is x = 16 inches and the possible error is 1 32 Step 2 Ax= dx = ± To approximate the possible propagated error in computing the area of the square, differentiate A = x² with respect to x. dA dx = where x is the side of the square. dA = x dx

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
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**Title: Error Approximation Using Differentials**

**Introduction:**

The side of a square floor tile is measured to be 16 inches, with a possible error of 1/32 inch. Use differentials to approximate the possible error and the relative error in computing the area of the square.

---

**Step 1:**

Recall that the formula for the area of a square is \( A = x^2 \), where \( x \) is the side of the square.

We are given that the side of the square floor tile is \( x = 16 \) inches and the possible error is 

\[
\Delta x = dx = \pm \frac{1}{32}.
\]

---

**Step 2:**

To approximate the possible propagated error in computing the area of the square, differentiate \( A = x^2 \) with respect to \( x \).

\[
\frac{dA}{dx} = \_\_\_ \cdot x
\]

\[
dA = \_\_\_ \cdot dx
\]

---

**Instructions:**

- Enter the derivative value in the text boxes provided.
- Click "Submit" once you have completed your calculations.

**Note:** Use this method to calculate the possible error (propagated and relative) in the area of the square to ensure precision in your measurements.

**Submit | Skip (you cannot come back)**

**Conclusion:**

Understanding and applying differentials allows us to handle measurement errors and their effects on calculations in practical scenarios.
Transcribed Image Text:**Title: Error Approximation Using Differentials** **Introduction:** The side of a square floor tile is measured to be 16 inches, with a possible error of 1/32 inch. Use differentials to approximate the possible error and the relative error in computing the area of the square. --- **Step 1:** Recall that the formula for the area of a square is \( A = x^2 \), where \( x \) is the side of the square. We are given that the side of the square floor tile is \( x = 16 \) inches and the possible error is \[ \Delta x = dx = \pm \frac{1}{32}. \] --- **Step 2:** To approximate the possible propagated error in computing the area of the square, differentiate \( A = x^2 \) with respect to \( x \). \[ \frac{dA}{dx} = \_\_\_ \cdot x \] \[ dA = \_\_\_ \cdot dx \] --- **Instructions:** - Enter the derivative value in the text boxes provided. - Click "Submit" once you have completed your calculations. **Note:** Use this method to calculate the possible error (propagated and relative) in the area of the square to ensure precision in your measurements. **Submit | Skip (you cannot come back)** **Conclusion:** Understanding and applying differentials allows us to handle measurement errors and their effects on calculations in practical scenarios.
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