Calculus Quiz: First Derivative of Parametric Curves

📚 Quick Study Guide

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• Parametric Equations: A set of equations that express a set of quantities as explicit functions of independent variables, known as "parameters." Often represented as $x = f(t)$ and $y = g(t)$.
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• First Derivative: The first derivative $\frac{dy}{dx}$ of a parametric curve defined by $x = f(t)$ and $y = g(t)$ is given by $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$, provided $\frac{dx}{dt} \neq 0$.
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• Chain Rule: Essential for finding the derivative. Remember that $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$.
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• Notation: $\frac{dx}{dt}$ represents the derivative of $x$ with respect to $t$, and $\frac{dy}{dt}$ represents the derivative of $y$ with respect to $t$.
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• Simplification: Always simplify your final answer to its simplest form.

🧪 Practice Quiz

Question 1: If $x = t^2$ and $y = 2t$, find $\frac{dy}{dx}$. A) $t$ B) $\frac{1}{t}$ C) $2t$ D) $\frac{1}{2t}$ Question 2: Given $x = \sin(t)$ and $y = \cos(t)$, determine $\frac{dy}{dx}$. A) $\tan(t)$ B) $-\tan(t)$ C) $\cot(t)$ D) $-\cot(t)$ Question 3: If $x = e^t$ and $y = t^2$, find $\frac{dy}{dx}$. A) $2te^{-t}$ B) $2te^{t}$ C) $t^2e^{-t}$ D) $t^2e^{t}$ Question 4: Let $x = t^3 + 1$ and $y = t^2 - 1$. Determine $\frac{dy}{dx}$. A) $\frac{2}{3t}$ B) $\frac{3t}{2}$ C) $\frac{2t}{3t^2}$ D) $\frac{3t^2}{2t}$ Question 5: If $x = \ln(t)$ and $y = \sqrt{t}$, find $\frac{dy}{dx}$. A) $\frac{1}{2\sqrt{t}}$ B) $\frac{\sqrt{t}}{2}$ C) $\frac{t}{2\sqrt{t}}$ D) $\frac{1}{2}t^{\frac{3}{2}}$ Question 6: Given $x = \cos(2t)$ and $y = \sin(2t)$, find $\frac{dy}{dx}$. A) $-\tan(2t)$ B) $\tan(2t)$ C) $-\cot(2t)$ D) $\cot(2t)$ Question 7: If $x = t + \frac{1}{t}$ and $y = t - \frac{1}{t}$, find $\frac{dy}{dx}$. A) $\frac{t^2 + 1}{t^2 - 1}$ B) $\frac{t^2 - 1}{t^2 + 1}$ C) $1$ D) $-1$

Click to see Answers

1. B) $\frac{1}{t}$
2. B) $-\tan(t)$
3. A) $2te^{-t}$
4. C) $\frac{2t}{3t^2}$
5. C) $\frac{t}{2\sqrt{t}}$
6. D) $\cot(2t)$
7. B) $\frac{t^2 - 1}{t^2 + 1}$