Question

use differentials to approximate the given value by hand.
$\sqrt{47.7}$

Explanation:

Step1: Choose the function and \(a\)

Let \(y = f(x)=\sqrt{x}\), and choose \(a = 49\) (since \(\sqrt{49}=7\) is easy to calculate).

Step2: Find the derivative of the function

The derivative of \(y = \sqrt{x}=x^{\frac{1}{2}}\) is \(y^\prime=f^\prime(x)=\frac{1}{2\sqrt{x}}\).

Step3: Calculate \(dx\)

\(x = 47.7\), \(a = 49\), so \(dx=x - a=47.7 - 49=- 1.3\).

Step4: Calculate \(dy\)

\(dy=f^\prime(a)dx\), since \(a = 49\) and \(f^\prime(x)=\frac{1}{2\sqrt{x}}\), then \(f^\prime(49)=\frac{1}{2\sqrt{49}}=\frac{1}{14}\).
\(dy=\frac{1}{14}\times(-1.3)=-\frac{1.3}{14}\approx - 0.0929\).

Step5: Approximate \(f(x)\)

\(f(x)\approx f(a)+dy\), \(f(49) = 7\), so \(\sqrt{47.7}\approx7-0.0929 = 6.9071\).

Answer:

\(\sqrt{47.7}\approx6.9071\)