Math 181  Midterm 2  November 9, 1992

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Show your work clearly and legibly. NO WORK – – NO CREDIT.

1. (20 pts) Sketch the graph of r = 2+sin 2θ, and find the slope of this curve at θ = \frac{π}{4} and at θ = \frac{π}{3} . [For θ = \frac{π}{3} you should leave \sqrt{3} in your answer. Recall that sin \frac{π}{3} = sin \frac{2π}{3} = \frac{\sqrt{3}}{2} and cos \frac{π}{3} = − cos \frac{2π}{3} = \frac{1}{2} and sin \frac{π}{4} = cos \frac{π}{4} = \frac{1}{\sqrt{2}}.]

Math 181  Midterm 2  November 9, 1992

2. (15 pts) Compute the area inside the cardioid r = 1 + cos θ.

3. (15 pts) Compute the length of the curve given by x = 2t^{3}, y = 3t^{2}, 0 ≤ t ≤ 1.

Math 181  Midterm 2  November 9, 1992

4. (15 pts) Determine whether the following series converges or diverges, explaining the test you are using. ∑_{n=1}^{∞} \frac{\sqrt{n^{4} − 1}}{2n^{3} + 1}

5. (15 pts) Write the recurring decimal 0.3518518˙51˙8 … as a fraction in its lowest form.

Math 181  Midterm 2  November 9, 1992

6. (15 pts) Compute ∫_{1}^{3} \sqrt{3 + 2x − x^{2}} dx