Applications of Derivatives
Parametric Equations
Grade 12
Question:
<p>For the parametric curve <span class='math'>x = x(t), y = y(t)</span>, find <span class='math'>\frac{dy}{dx}</span> at the point corresponding to <span class='math'>t = 2</span>, given that at this point <span class='math'>\frac{dy}{dt} = 4t - 2</span> and <span class='math'>\frac{dx}{dt} = 2t + 3</span>.</p>
Step-by-Step Solution
Key Concept: Use the parametric derivative formula: the derivative of y with respect to x equals the ratio of derivatives with respect to the parameter.
<p><strong>Solution:</strong> For parametric equations, <span class='math'>\frac{dy}{dx} = \frac{dy/dt}{dx/dt}</span></p><p>At <span class='math'>t = 2</span>:</p><p><span class='math'>\frac{dy}{dt}\bigg|_{t=2} = 4(2) - 2 = 6</span></p><p><span class='math'>\frac{dx}{dt}\bigg|_{t=2} = 2(2) + 3 = 7</span></p><p>Therefore, <span class='math'>\frac{dy}{dx}\bigg|_{t=2} = \frac{6}{7}</span></p>
Correct Answer: 6/7