Applications of Derivatives
Tangents to curves
Grade 12
Question:
<p>For the curve \(y = 3\sin\theta\cos\theta\), \(x = e^\theta\sin\theta\), \(0 \le \theta \le \pi\), the tangent is parallel to x-axis when \(\theta\) is</p>
<p>\(\dfrac{3\pi}{4}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{\pi}{6}\)</p>
<p>\(\dfrac{260}{\sqrt{37}}\)</p>
Step-by-Step Solution
Key Concept: The tangent is parallel to the x-axis when dy/dx = 0, which occurs when dy/dθ = 0 while dx/dθ ≠ 0 (for parametric curves). We must also verify the tangent isn't vertical at that point.
<p><strong>Step 1:</strong> Find dy/dθ and dx/dθ</p><p>Given: y = 3sin θ cos θ = (3/2)sin(2θ)</p><p>dy/dθ = 3(cos²θ - sin²θ) = 3cos(2θ)</p><p></p><p>For x = e^θ sin θ (using product rule):</p><p>dx/dθ = e^θ sin θ + e^θ cos θ = e^θ(sin θ + cos θ)</p><p></p><p><strong>Step 2:</strong> Set dy/dx = 0</p><p>For tangent parallel to x-axis: dy/dx = 0</p><p>This requires: dy/dθ = 0 and dx/dθ ≠ 0</p><p></p><p>3cos(2θ) = 0</p><p>cos(2θ) = 0</p><p>2θ = π/2, 3π/2</p><p>θ = π/4, 3π/4</p><p></p><p><strong>Step 3:</strong> Verify dx/dθ ≠ 0</p><p>At θ = π/4: e^(π/4)(sin(π/4) + cos(π/4)) = e^(π/4)·√2 ≠ 0 ✓</p><p>At θ = 3π/4: e^(3π/4)(sin(3π/4) + cos(3π/4)) = e^(3π/4)·0 = 0 ✗</p><p></p><p>∴ Answer: B (θ = π/4)</p>
Correct Answer: B