Applications of Derivatives
Tangent to a curve
Grade 12
Question:
<p>The tangent to the curve \(y = xe^{x^2}\) at the point \((1, e)\), also passes through the point:</p>
<p>\(\left(\dfrac{5}{3}, 2e\right)\)</p>
<p>\(\left(\dfrac{4}{3}, 2e\right)\)</p>
<p>\((3, 6e)\)</p>
<p>\((2, 3e)\)</p>
Step-by-Step Solution
Key Concept: Find the tangent line equation using the derivative at the given point, then check which point satisfies this linear equation.
<p><strong>Step 1:</strong> Find the derivative of y = xe^(x²).</p><p>Using the product rule: dy/dx = e^(x²) + x·e^(x²)·2x = e^(x²)(1 + 2x²)</p><p><strong>Step 2:</strong> Evaluate the slope at point (1, e).</p><p>At x = 1: dy/dx = e^(1)(1 + 2) = 3e</p><p><strong>Step 3:</strong> Write the equation of the tangent line using point-slope form.</p><p>y - e = 3e(x - 1)</p><p>y = 3ex - 3e + e</p><p>y = 3ex - 2e</p><p><strong>Step 4:</strong> Check which of the given options satisfies this equation.</p><p>Substitute the coordinates of each option into y = 3ex - 2e and verify which point lies on this line.</p><p>∴ Answer: B</p>
Correct Answer: B