<p>The area (in sq. units) bounded by the parabola \(y = x^2 - 1\), the tangent at the point (2, 3) to it and the \(y\)-axis is __________ (up to three decimal places).</p>
Step-by-Step Solution
Key Concept: Find the tangent line equation at (2,3), then integrate to find the area between the parabola, tangent line, and y-axis by identifying which curve is above the other in the relevant region.
<p><strong>Step 1:</strong> Find the tangent line at point (2, 3).</p><p>For y = x² - 1, we have dy/dx = 2x</p><p>At x = 2: dy/dx = 4</p><p>Tangent line: y - 3 = 4(x - 2) → y = 4x - 5</p><p><strong>Step 2:</strong> Find where tangent meets the y-axis.</p><p>At x = 0: y = -5</p><p>So tangent intersects y-axis at (0, -5)</p><p><strong>Step 3:</strong> Identify the region and set up the integral.</p><p>At x = 0: parabola gives y = -1, tangent gives y = -5</p><p>At x = 2: both give y = 3</p><p>For 0 ≤ x ≤ 2, the parabola y = x² - 1 is above the tangent y = 4x - 5</p><p><strong>Step 4:</strong> Calculate the area.</p><p>A = ∫₀² [(x² - 1) - (4x - 5)] dx</p><p>A = ∫₀² [x² - 4x + 4] dx</p><p>A = ∫₀² [(x - 2)²] dx</p><p>A = [(x - 2)³/3]₀²</p><p>A = [0 - (-8/3)] = 8/3</p><p>∴ Answer: <strong>2.667</strong> sq. units</p>
Correct Answer: 2