Applications of Derivatives
Approximations and Errors
Grade 12
Question:
<p>If <strong>f</strong>(<strong>x</strong>) = 3<strong>x</strong>² + 15<strong>x</strong> + 5, then the approximate value of <strong>f</strong>(3.02) is</p>
<p>(a) 47.66</p>
<p>(b) 57.66</p>
<p>(c) 67.66</p>
<p>(d) 77.66</p>
Step-by-Step Solution
Key Concept: Linear approximation using differentials: when Δx is small, Δy ≈ dy = f'(x)Δx allows us to approximate function values without direct calculation.
<p><strong>Step 1:</strong> Given <strong>f</strong>(<strong>x</strong>) = 3<strong>x</strong>² + 15<strong>x</strong> + 5</p><p>Therefore, <strong>f</strong>'(<strong>x</strong>) = 6<strong>x</strong> + 15</p><p><strong>Step 2:</strong> Let <strong>x</strong> = 3 and Δ<strong>x</strong> = 0.02</p><p><strong>Step 3:</strong> Use the approximation formula: <strong>f</strong>(<strong>x</strong> + Δ<strong>x</strong>) ≈ <strong>f</strong>(<strong>x</strong>) + Δ<strong>x</strong> · <strong>f</strong>'(<strong>x</strong>)</p><p><strong>Step 4:</strong> <strong>f</strong>(3.02) ≈ [3(3)² + 15(3) + 5] + (0.02)[6(3) + 15]</p><p>= [27 + 45 + 5] + (0.02)[18 + 15]</p><p>= 77 + (0.02)(33)</p><p>= 77 + 0.66</p><p>= 77.66</p><p>∴ Answer is (d) 77.66</p>
Correct Answer: D