Parametric, polar and vector functions Knowledge-Based Practice

Questions

1. 1.

   What is the total length of the polar curve $r = e^{2\theta}$ on the interval $0 \le \theta \le \pi$?

   • A) $\frac{\sqrt{3}}{2}(e^{2\pi} - 1)$
   • B) $\sqrt{5}(e^{2\pi} - 1)$
   • C) $\frac{\sqrt{5}}{2}(e^{2\pi} - 1)$
   • D) $\frac{5}{2}(e^{2\pi} - 1)$
2. 2.

   What is the slope of the line tangent to the polar curve $r = 2 + \cos\theta$ at $\theta = \frac{\pi}{2}$?

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

   What is the area of the region enclosed by one loop of the polar curve $r = 6\sin(3\theta)$?

   • A) $\frac{3\pi}{2}$
   • B) $\frac{9\pi}{2}$
   • C) $3\pi$
   • D) $6\pi$
4. 4.

   A particle moves in the $xy$-plane such that its velocity vector is given by $\vec{v}(t) = \langle 6t, 3t^2 - 3 \rangle$ for $t \ge 0$. What is the total distance traveled by the particle from $t = 0$ to $t = 2$?

   • A) 10
   • B) 12
   • C) 14
   • D) 18
5. 5.

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

   • A) $-\frac{10}{81}$
   • B) $-\frac{10}{9}$
   • C) $\frac{1}{2}$
   • D) $\frac{10}{81}$

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