Parametric functions Knowledge-Based Practice

Questions

1. 1.

   What is the arc length of the curve defined parametrically by $x(t) = \frac{1}{3}t^3$ and $y(t) = \frac{1}{2}t^2$ for $0 \le t \le \sqrt{3}$?

   • A) $\frac{7}{3}$
   • B) $\frac{14}{3}$
   • C) $\frac{8}{3}$
   • D) $\frac{7}{2}$
2. 2.

   A curve in the $xy$-plane is defined parametrically by $x(t) = t^2 - 1$ and $y(t) = t^3 - 3t$. What is the value of $\frac{d^2y}{dx^2}$ at the point where $t = 2$?

   • A) $6$
   • B) $\frac{15}{8}$
   • C) $\frac{9}{32}$
   • D) $\frac{15}{32}$
3. 3.

   A curve in the $xy$-plane is defined by the parametric equations $x(t) = 3 + 2\ln(t)$ and $y(t) = t^2 - 1$ for $t > 0$. Which of the following is the Cartesian equation of this curve?

   • A) $y = e^{2(x - 3)} - 1$
   • B) $y = \frac{1}{4}(x - 3)^2 - 1$
   • C) $y = e^{x - 3} - 1$
   • D) $y = e^{\frac{x - 3}{2}} - 1$
4. 4.

   A curve in the $xy$-plane is defined parametrically by $x(t) = 4\cos(t) - 1$ and $y(t) = 3\sin(t) + 2$ for $0 \le t \le 2\pi$. Which of the following is the Cartesian equation of this curve?

   • A) $\frac{(x-1)^2}{16} + \frac{(y+2)^2}{9} = 1$
   • B) $\frac{(x+1)^2}{16} + \frac{(y-2)^2}{9} = 1$
   • C) $\frac{(x+1)^2}{9} + \frac{(y-2)^2}{16} = 1$
   • D) $\frac{(x-1)^2}{9} + \frac{(y+2)^2}{16} = 1$
5. 5.

   A curve is defined parametrically by the equations $x(t) = 2t^3 - 3t$ and $y(t) = t^4 + 2t$. What is the slope of the tangent line to the curve at the point where $t = 1$?

   • A) 2
   • B) $\frac{1}{2}$
   • C) 3
   • D) 6

Answers and step-by-step solutions appear after you finish the test.