Use the differential to \approx imate the radical expression. Then use a calculator to \approx imate the quantity, and give the absolute value of the difference in the two results to 4 decimal places. squ

To use the differential to \approx imate the radical expression {eq}sqrt{51} {/eq}, we can use the first-order Taylor expansion: {eq}sqrt{51 + Delta x} \approx sqrt{51} + \frac{1}{2sqrt{51}}Delta x {/eq} where {eq}Delta x {/eq} is a small change in the input. We can choose {eq}Delta x = 1 {/eq} for convenience, which gives: {eq}sqrt{52} \approx sqrt{51} + \frac{1}{2sqrt{51}} \approx 7.14142842854 {/eq} Using a calculator, we obtain: {eq}sqrt{51} \approx 7.14142842854 {/eq} The absolute value of the difference between the two results is: {eq}|7.14142842854 - 7.14142842854| = 0 {/eq} rounded to 4 decimal places. This is expected since we used the first-order Taylor expansion, which is accurate to first-order in {eq}Delta x {/eq}. The actual error may be higher if we use a larger value of {eq}Delta x {/eq} or higher-order Taylor expansions.