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Workshop #12, Math 143

Today’s Topics:

● Polar arc length and area

● Final exam review

Warm-up: Consider the polar curve T = θ . Draw a quick sketch the curve, and ﬁnd the length of the curve for 0 · θ · 1.

Workshop questions:

1. Consider the polar curve T = 4 sin(2θ). (a) Sketch a graph of the curve.

m/2

3m/2

(b) Find the area enclosed by one loop of the curve.

2. Consider the polar curve T = 3 - 2 cos(θ). Sketch a graph of the curve, and ﬁnd the total area enclosed by the curve.

m/2

3m/2

3. Consider the polar curve T = 3 cos(θ) - 2. (a) Sketch the curve.

m/2

3m/2

(b) Find the angles where the curve crosses itself.

(d) Find the area between the outer loop and the inner loop of the curve.

Final Exam Review:

4. Determine whether the following sequences converge or diverge. If they converge, ﬁnd their limit. If they diverge, state whether they diverge to +● , -● or because they oscillate. Justify and show all your work.

(a) an = tan → 1 ╱ 、

(b) an = tan → 1 ╱ 、

(d) an = ╱ 、n (e) an = ╱ 、n (f) an = ╱ 、n (g) an = (-1)n

(h) an =

(i) an =

(j) an =

(k) an =

5. Determine the convergence/divergence behavior of each series using as many diﬀerent tests as possible. (v) → → 1 n2 + 1 n=1 (x) → arctan(n) → → 1 n! n=1 (z) → (t) 4n n=1 (u) 4n n=1
6. Determine whether the following series converge absolutely, converge only conditionally, or diverge. Name any test you use and justify its use.
(a) → ^n cos(n) n=1 → (-1)n 4n n=1	→ (-1)n n ln(n) n=1 (d) →	(e) → ╱ 、n → (-1)n n!
7. Now make up a series that converges absolutely, one that converges only conditionally, and one that diverges.
8. Write down your approach to general series testing. What steps do you take? What do you look for ﬁrst? Last? What test(s) do you start with? What kinds of series do you use which tests on? Make up a series that is telescoping; make up a series on which you’d use the test for divergence, alternating series test, p-test, geometric series test, comparison test, limit comparison test, and root/ratio test.
→ 9. Find the radius and interval of convergence of the following power series: n=1			(-1)n n5n	(5x - 2)n .

10. Find a Maclaurin series for f (x) = , and its radius and interval of convergence.

11. Consider the function f (x) = sin(5x). Find a power series expansion of f (x) about x = . Write out the ﬁrst three nonzero terms, and express the series in sigma notation. Use the ratio test to ﬁnd the radius and interval of convergence of the series you found above.