Applications of Derivatives
Approximate Values Using Differentials
Grade 12
Question:
<p>66. (A) In a circle, \(r = 5\) cm and \(\Delta r = 0.06\) cm. Find the approximate change in area \(\Delta A\) using differentials, where \(A = \pi r^2\).</p>
Step-by-Step Solution
Key Concept: Use differentials to approximate changes in area; $dA = 2\pi r \, dr$ gives the linear approximation.
<p><strong>Step 1:</strong> We have $A = \pi r^2$</p><p><strong>Step 2:</strong> Taking differential: $dA = 2\pi r \, dr$</p><p><strong>Step 3:</strong> Substitute $r = 5$ and $dr = 0.06$:</p><p>$dA = 2\pi(5)(0.06) = 0.6\pi$ cm²</p><p>∴ Approximate change in area is $0.6\pi$ cm².</p>
Correct Answer: 0.6π