Calculus review: Eliminating the parameter multiple choice questions

📚 Quick Study Guide

• 🧭 Parametric Equations: Equations where $x$ and $y$ are defined in terms of a third variable, usually $t$. Example: $x = f(t)$, $y = g(t)$.
• ✏️ Eliminating the Parameter: The process of rewriting parametric equations as a single equation in terms of $x$ and $y$.
• 💡 Common Techniques:
  • Solve one equation for $t$ and substitute into the other.
  • Use trigonometric identities if trigonometric functions are involved (e.g., $\sin^2(t) + \cos^2(t) = 1$).
  • Consider algebraic manipulation to isolate $t$ terms.
• 📝 Important Trigonometric Identities:
  • $\sin^2(\theta) + \cos^2(\theta) = 1$
  • $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Practice Quiz

1. What is the Cartesian equation for the parametric equations $x = t + 1$ and $y = t^2$?
   1. $y = (x - 1)^2$
   2. $y = x^2 - 1$
   3. $y = x^2 + 1$
   4. $y = x^2$
2. What is the Cartesian equation for the parametric equations $x = 2\cos(\theta)$ and $y = 2\sin(\theta)$?
   1. $x^2 + y^2 = 2$
   2. $x^2 + y^2 = 4$
   3. $\frac{x^2}{4} + \frac{y^2}{4} = 1$
   4. $\frac{x^2}{2} + \frac{y^2}{2} = 1$
3. Eliminate the parameter $t$ from $x = t^3$ and $y = t^6 + 1$ to find the Cartesian equation.
   1. $y = x^2 + 1$
   2. $y = x^3 + 1$
   3. $y = x^2 - 1$
   4. $y = x^6 + 1$
4. Find the Cartesian equation corresponding to the parametric equations $x = \sqrt{t}$ and $y = t + 3$.
   1. $y = x^2 + 3$
   2. $y = x + 3$
   3. $y = x^4 + 3$
   4. $y = \sqrt{x} + 3$
5. Which Cartesian equation is equivalent to $x = e^t$ and $y = e^{2t}$?
   1. $y = x$
   2. $y = x^2$
   3. $y = 2x$
   4. $y = e^{x}$
6. Eliminate the parameter $t$ in the equations $x = t - 1$ and $y = \frac{1}{t - 1}$.
   1. $y = \frac{1}{x}$
   2. $y = \frac{1}{x + 1}$
   3. $y = x - 1$
   4. $y = \frac{1}{x - 1}$
7. Determine the Cartesian form of the parametric equations $x = 3\sin(t)$ and $y = 5\cos(t)$.
   1. $\frac{x^2}{9} + \frac{y^2}{25} = 1$
   2. $\frac{x^2}{25} + \frac{y^2}{9} = 1$
   3. $9x^2 + 25y^2 = 1$
   4. $25x^2 + 9y^2 = 1$