Warm-Up.
Answer each of the following as best you can. Feel free to discuss your thoughts with peers around you.
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Worksheet Session #12: Final Exam Review Part 2 -- Chapters 10 & 11
1.
To find the Cartesian coordinates \((x, y)\) when the polar coordinates \((r, \theta)\) are known, which equations should be used?
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\(\displaystyle x = r\sin(\theta),\ y = r\cos(\theta)\)
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\(\displaystyle r = x\cos(\theta),\ r = y\sin(\theta)\)
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\(\displaystyle r^2 = x^2 + y^2,\ \tan(\theta) = \dfrac{y}{x}\)
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\(\displaystyle x = r^2\cos(\theta),\ y = r^2\sin(\theta)\)
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\(\displaystyle x = r\cos(\theta),\ y = r\sin(\theta)\)
2.
The Cartesian equation for the polar curve \(r^2 = 10\) is .
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\(\displaystyle (x - 10)^2 + y^2 = 1\)
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\(\displaystyle x^2 + (y + 10)^2 = 1\)
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\(\displaystyle x^2 + y^2 = 10\)
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\(\displaystyle (x - 10)^2 + (y + 10)^2 = 1\)
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\(\displaystyle 10x^2 + 10y^2 = 1\)
3.
The Cartesian equation for the polar curve \(r^2\sin(2\theta) = 1\) is .
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\(\displaystyle y = \dfrac{x}{2}\)
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\(\displaystyle y = 2x\)
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\(\displaystyle 2x^2 + y^2 = 1\)
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\(\displaystyle y = \dfrac{1}{x}\)
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\(\displaystyle y = \dfrac{1}{2x}\)
Directions: Answer each of the following with “true” or “false”.
4.
If \(\displaystyle\lim_{n\to\infty} a_n = 0\text{,}\) then \(\displaystyle\sum_{n=1}^{\infty} a_n\) is convergent.
5.
If \(\displaystyle\sum_{n=1}^{\infty} a_n\) is convergent, then \(\displaystyle\lim_{n\to\infty} a_n = 0\text{.}\)
6.
The series \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^p}\) is convergent if \(p \geq 1\) and divergent if \(p \lt 1\text{.}\)
Hands-On.
Answer each of the following as best you can. Feel free to discuss your thoughts with peers around you.
1.
Given the Cartesian point \((-4, 4)\text{,}\) find the polar coordinates
(a)
when \(r \gt 0\text{.}\)
(b)
when \(r \lt 0\text{.}\)
2.
Find the area of the region enclosed by one loop of the curve \(r = 4\cos(3\theta)\text{.}\)
3.
Find the area of the region inside the curve \(r = 3\cos(\theta)\) and outside the curve \(r = 1 + \cos(\theta)\text{.}\)
4.
5.
Find the Maclaurin series for the function \(f(x) = x^2 e^{-x^3}\text{.}\)
6.
At what point(s) on the curve \(x = t^2 + 1,\ y = t^3 - 1\) does the tangent line have slope \(\dfrac{1}{2}\text{?}\)
7.
Determine whether the following series converge or diverge. Justify your answers.
(a)
\(\displaystyle\sum_{n=1}^{\infty} \frac{\sin(n) + 5}{n^{3/2}}\)
(b)
\(\displaystyle\sum_{n=1}^{\infty} \frac{5n \cdot e^{2n}}{4^n}\)
(c)
\(\displaystyle\sum_{n=2}^{\infty} \frac{\sqrt{n} + 4}{n^3 - 3n}\)
8.
(a)
Use the integral test to show that the series \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^3}\) converges.
(b)
How many terms are needed to guarantee that the \(n\)-th partial sum \(s_n\) is accurate to within \(\dfrac{1}{100}\text{?}\)
9.
Consider the curve \(x = \cos(t),\ y = 3 + \sin(t)\) for \(0 \leq t \leq 2\pi\text{.}\) Find the area of the surface generated by revolving this curve around the \(x\)-axis.
10.
Find the length of the curve \(\displaystyle x = \frac{1}{3}(2t + 3)^{3/2},\ y = t + \frac{t^2}{2}\) for \(0 \leq t \leq 2\text{.}\)
Grades Up!
Choose the best answer for each of the following.
1.
Eliminating the parameter from the curve \(x = 2\cos(t),\ y = 3\sin(t)\) for \(0 \leq t \leq 2\pi\) gives the Cartesian equation .
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\(\displaystyle \dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\)
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\(\displaystyle \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1\)
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\(\displaystyle x^2 + y^2 = 1\)
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\(\displaystyle x^2 + y^2 = 13\)
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None of these.
2.
Which of the following is true for \(\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^n}{n\,\ln(n)}\text{?}\)
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The series converges absolutely.
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The series converges, but not absolutely.
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The series diverges by the Alternating Series Test.
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The series diverges by the Test for Divergence.
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None of these.
3.
Find the sum of the series \(\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n\,3^{2n+1}}{5^{2n}\,(2n)!}\text{.}\)
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\(\displaystyle 3\sin\!\left(\dfrac{3}{5}\right)\)
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\(\displaystyle -3\cos\!\left(\dfrac{3}{5}\right)\)
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\(\displaystyle -3\sin\!\left(\dfrac{3}{5}\right)\)
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\(\displaystyle 3\cos\!\left(\dfrac{3}{5}\right)\)
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\(\displaystyle -\sin\!\left(\dfrac{3}{5}\right)\)
4.
The series \(\displaystyle\sum_{n=1}^{\infty} \frac{3^n}{(n+1)!}\)
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diverges by the integral test.
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converges by the ratio test.
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converges by the integral test.
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diverges by the ratio test.
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diverges by the test for divergence.
5.
The interval of convergence for \(\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^n (3x - 1)^n}{5^n (n - 1)}\) is .
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\(\displaystyle \left(-\dfrac{4}{3},\ 2\right)\)
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\(\displaystyle \left[-\dfrac{4}{3},\ 2\right]\)
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\(\displaystyle \left(-\dfrac{4}{3},\ 2\right]\)
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\(\displaystyle \left(0,\ \dfrac{1}{3}\right]\)
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\(\displaystyle \left(0,\ \dfrac{1}{3}\right)\)
6.
Which integral will correctly calculate the surface area of the object obtained by rotating the curve \(y = e^{-x/2}\text{,}\) \(0 \leq x \lt 2\) about the \(x\)-axis?
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\(\displaystyle \displaystyle\int_0^2 e^{-t/2}\sqrt{1 + \tfrac{1}{4}e^{-t}}\,dt\)
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\(\displaystyle \displaystyle 2\pi\int_0^2 e^{-x}\sqrt{1 + \tfrac{1}{4}e^{-x}}\,dx\)
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\(\displaystyle \displaystyle 2\pi\int_0^2 e^{-t/2}\sqrt{\tfrac{1}{4}e^{-t}}\,dt\)
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\(\displaystyle \displaystyle 2\pi\int_0^2 e^{-t/2}\left(1 + \tfrac{1}{4}e^{-t}\right)^{2}\,dt\)
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\(\displaystyle \displaystyle 2\pi\int_0^2 e^{-t/2}\sqrt{1 + \tfrac{1}{4}e^{-t}}\,dt\)
