Math 243: Calculus II


Unit IV Exam: Review Guide


The Unit IV Exam will be divided into a take-home portion and an in-class portion. The take-home portion was handed out in class and can be found on the Exams page.

This page contains problems similar to those that will appear on the in-class portion of the Unit IV Exam. These questions are all similar to ones that you encountered while doing your WebAssign homework.


1. Find polar equations for the curves represented by the Cartesian equations.
a.) \(x + y = 2\)
b.) \(x^2 + y^2 = 2\)

2. Sketch the polar curves.
a.) \( r = \cos(3 \theta) \)
b.) \( r = 3\cos\theta \)
c.) \( r = 3 + \cos\theta \)

3. Consider the ellipse given by the polar equation \[ r = \dfrac{2}{4 - 3\cos\theta}.\] Sketch the graph of the ellipse and compute its perimeter and area.

4. Find the slope of the tangent line to the parametric curve \[\begin{cases} x = \ln t, \\[1.5 ex] y = 1 + t^2 \end{cases}\] at the point \(t = 1\).

5. Find \(\tfrac{dy}{dx}\) and \(\tfrac{d^2y}{dx^2}\) for the parametric curve \[\begin{cases} x = t + \sin t, \\[1.5 ex] y = t - \cos t. \end{cases}\]

6. Find the area enclosed by the inner loop of the curve \(r = 1 - 3\sin\theta\).

7. Find the length of the curve \[\begin{cases} x = 2 + 3t, \\[1.5 ex] y = \cosh(3t) \end{cases}\] on the interval \(0\leq t \leq 1\).

8. Find the area of the surface obtained by rotating the curve segment \[\begin{cases} x = 2 + 3t, \\[1.5 ex] y = \cosh(3t), \\[1.5 ex] 0\leq t \leq 1 \end{cases}\] about the \(x\)-axis.

9. Find the area of the surface obtained by rotating the curve segment \[\begin{cases} x = 2 + 3t, \\[1.5 ex] y = \cosh(3t), \\[1.5 ex] 0\leq t \leq 1 \end{cases}\] about the \(y\)-axis. [At least set the integral up by hand, then use a computer to evaluate it, if necessary.]

10. Find a polar equation for the parabola with focus at the origin and directrix with equation \(r = 4\sec\theta\).

11. Find a polar equation for the ellipse with focus at the origin, eccentricity \(e = \tfrac{1}{3}\), and directrix with equation \(r = 4 \sec\theta\).

12. Find a polar equation for the hyperbola with focus at the origin, eccentricity \(e = 3\), and directrix with equation \(r = 4 \sec\theta\).

13. Plot the conic sections of questions 10 through 12 on the same set of axes, together with their common focus on and directrix.




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