(C) \(\cot^2 \theta\)
(D) \(\sin^4 \theta\)
What is the average rate of change of \(f(x) = \sin(x)\) on the interval \([0, \pi]\)?
(A) \(0\)
(B) \(\frac{2}{\pi}\)
(C) \(1\)
(D) \(\frac{\pi}{2}\)
Solve the equation \(2\cos(x) + 1 = 0\) for \(0 \leq x < 2\pi\).
(A) \(x = \frac{\pi}{3}, \frac{5\pi}{3}\)
(B) \(x = \frac{2\pi}{3}, \frac{4\pi}{3}\)
(C) \(x = \frac{\pi}{6}, \frac{11\pi}{6}\)
(D) \(x = \frac{5\pi}{6}, \frac{7\pi}{6}\)
Find the rectangular coordinates for the polar point \((4, \frac{7\pi}{6})\).
(A) \((2\sqrt{3}, 2)\)
(B) \((-2, -2\sqrt{3})\)
(C) \((-2\sqrt{3}, -2)\)
(D) \((2, -2\sqrt{3})\)
Convert the rectangular equation \(x^2 + y^2 = 6x\) to a polar equation.
(A) \(r = 6\)
(B) \(r = 6\cos \theta\)
(C) \(r = 6\sin \theta\)
(D) \(r^2 = 6\cos \theta\)
The polar graph \(r = 4\sin(3\theta)\) is a rose curve. How many petals does it have?
(A) 3
(B) 4
(C) 6
(D) 8
Which of the following describes the graph of \(r = 2 + 3\cos \theta\)?
(A) A cardioid
(B) A limaçon with an inner loop
(C) A dimpled limaçon
(D) A circle
A parametric function is defined by \(x(t) = 2t + 1\) and \(y(t) = t^2 - 3\). Which rectangular equation represents this path?
(A) \(y = (\frac{x-1}{2})^2 - 3\)
(B) \(y = (2x + 1)^2 - 3\)
(C) \(x = 2y^2 - 3\)
(D) \(y = 4x^2 - 3\)
Which function has a range of \([0, \pi]\)?
(A) \(y = \arcsin(x)\)
(B) \(y = \arccos(x)\)
(C) \(y = \arctan(x)\)
(D) \(y = \sin(x)\)
The height of tide in a harbor is modeled by \(h(t) = 4\sin(\frac{\pi}{6}(t - 2)) + 10\), where \(t\) is hours after midnight. What is the frequency of the tide in cycles per hour?
(A) \(\frac{\pi}{6}\)
(B) 12
(C) \(\frac{1}{12}\)
(D) 6
What is the rate of change of \(r = 2 + 2\cos \theta\) with respect to \(\theta\) at \(\theta = \frac{\pi}{2}\)?
(A) 0
(B) -2
(C) 2
(D) -1
The position of a particle at time \(t\) is given by \(x(t) = \cos(t)\) and \(y(t) = 3\sin(t) - 1\). What is the shape of the path traced by the particle?
(A) A circle
(B) A horizontal ellipse
(C) A vertical ellipse
(D) A hyperbola
Which of the following intervals for \(x\) would make \(f(x) = \sin(x)\) invertible?
(A) \([0, \pi]\)
(B) \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
(C) \([\pi, 2\pi]\)
(D) \([0, 2\pi]\)
17. Frequency: \(P = 12\) hours. Frequency \(= 1/P = 1/12\) cycles/hour.
18. Rate: \(\frac{dr}{d\theta} = -2\sin\theta\). At \(\pi/2\), \(-2\sin(\pi/2) = -2\).
19. Path: \(x^2 + (\frac{y+1}{3})^2 = 1\). This is an ellipse with a longer vertical axis.
20. Restricted Domain: \(\sin(x)\) is one-to-one on \([-\pi/2, \frac{\pi}{2}]\).
Scoring Guidelines
5 pts per correct answer. 100 points total.
