Integration

This is one of the best reviews I have seen. All required skills are present, and there is a good transition from easy to hard for each category. Well done! There are a couple of blunders (integrals labeled as belonging to one technique that are not solvable that way, for example), but in general it does not affect too much to the overall quality. My selection of improper integrals may be a bit more interesting than yours, though. A solid 5/5, nevertheless.

1. Compute the following integrals:
   \begin{align} &\int \cos^3 \big(\tfrac{x}{2}\big) \csc^2 \big( \tfrac{x}{2}\big) \, dx &&\int p(p+5)^8\, dp && \int x^3\sqrt{x^2-6}\, dx \\ &\int e^x \cos x\, dx && \int x^3e^{2x}\, dx &&\int x^2\sin x\, dx \\ &\int \sin^3 x\cos^2 x\, dx && \int x\sin^{-1}(x^2)\, dx &&\int e^x\sec^2 x\, dx \\ &\int 6\cos^3 \big(\tfrac{x}{10}\big)\, dx &&\int 4\cos^2 (2\theta) \sin (2\theta)\, d\theta && \int 10\sin^2 (5x) \cos^2 (5x)\, dx \\ &\int \frac{2x^2}{\sqrt{16-x^2}}\, dx && \int \frac{\sqrt{9x^2-1}}{x^3}\, dx &&\int \frac{dx}{x^2\sqrt{x^2+1}} \\ & \int \frac{x^2-7x+3}{x+3}\, dx &&\int \frac{x^3-4}{x-2}\, dx && \int \frac{3x^2+5x-3}{x-1}\, dx \\ &\int \frac{6x+22}{2(x^2-x-6)}\, dx && \int \frac{2x^3+x^2+2x+8}{(x^2+1)^2}\, dx &&\int \frac{11x+81}{(x+7)(x+9)}\, dx \\ &\int \frac{x+1}{x^2(x-1)}\, dx &&\int \frac{x-6}{x^2(x+1)}\, dx && \int x\ln x\, dx \\ \end{align}
2. Compute the following definite integrals:
   \begin{align} &\int_0^2 \frac{dx}{\sqrt{4-x^2}} &&\int_{-\infty}^6 \frac{6}{x^2+36}\, dx \end{align}
3. Compute the derivative of the following functions:
   \begin{align} F(x) &= \int_x^5 \sqrt{1+t^4}\, dt \\ F(x) &= \int_0^{3x^2-2x+e^x} \sqrt{1+t^2}\, dt \end{align}

Sequences, Series

These problems are a lot of fun! You guys have done an excellent job. There are seriously evil problems in that collection. There were a couple of issues: 1. Two students presented problems from the same topic. Only one of them will get the points, and the corresponding problems featured here. 2. The student in charge of sequences/partial sums completely missed the point. Watch out for those questions in the final! I always make sure to include at least one or two problems in which you have to find the general term of a sequence, or the limit of a sequence, or a partial sum of a sequence. Please, do refer to the HW or the tests for that part. Anyway, the selection of series was so much fun that I ended up giving y'all a 4/5.

1. Address the convergence of the following series. Indicate what test you have used:
   \begin{align} &\sum_{n=0}^\infty \frac{n^7}{5n^7+3} && \sum_{n=0}^\infty \frac{1}{2^n} && \sum_{n=1}^\infty (-1)^n \frac{1}{n^3} \\ &\sum_{n=2}^\infty \frac{5n}{n^3+7n+13} && \sum_{n=2}^\infty \frac{4n^2}{\sqrt{9n^2+3}} && \sum_{n=1}^\infty \bigg( \frac{2n+5}{5n-1} \bigg)^n \\ &\sum_{n=1} \frac{(-5)^n}{n^2 2^n} && \sum_{n=0}^\infty n!(-e)^{-n} && \sum_{n=1}^\infty (-1)^n \frac{n^2(n+2)!}{n!3^{2n}} \end{align}
2. Compute the sum of the following series:
   \begin{align} &\sum_{n=1}^\infty \frac{1}{n^2+n} && \sum_{n=0} \big( \tan^{-1}n - \tan^{-1}(n+1) \big) \\ &\sum_{n=1}^\infty 5 \big( \tfrac{3}{4} \big)^{n-1} && \sum_{n=10}^\infty \frac{1}{8^n} \end{align}

Power series

Very weak! Some of the questions didn't make sense (I did not include those in this selection), some of you managed to write exactly the same question in different categories, some of you wrote very simple questions that don't test any of the required skills... This is not very representative of what will be in the exam. Refer to our fourth test instead, and the HW online for reference. This test receives a 2/5 (mostly for the good effort of a few).

1. Find a power series representation of the following functions (and include the center and radius):
   \begin{align} f(x) &= \frac{5}{3-x} &f(x) &= \frac{3}{x-2} & f(x) &= \frac{1}{1+x^4} \\ f(x) &= x^{-2} &f(x) &= \tan^{-1} x & f(x) &= \ln (1-x) \text{ for } \tfrac{1}{2} \leq x \leq 2 \\ f(x) &= e^{-x} &f(x) &= 4\sin(x^4) & f(x) &= 3\cos(3x) \end{align}
2. Find center, radius, and interval of convergence of the following power series:
   \begin{align} &\sum_{n=1}^\infty \frac{(x-1)^{n}}{4n} &&\sum (-1)^n (4x+1)^n \\ &\sum_{n=0}^\infty \frac{n+1}{(2n+1)!} (x-2)^n && \sum_{n=0}^\infty \frac{4^{1+2n}}{5^{n+1}}(x+3)^n \\ \end{align}

3. Find the Taylor series of the function \( f(x) = 2x^2+2x+1 \) centered around \( b=2. \)

4. Find a Taylor polynomial of degree 2 generated by \( f(x) = \sin x \) around \( b=\pi/4. \)

5. Find a MacLaurin series representation for the function \( f(x) = x^6 - e^{2x^3}. \)

Parametric and Polar equations

All the different skills seem to be there, except the computation of lengths of curves given in parametric. Make sure to study that part too. Most of the problems are moderate-to-easy. Remember that in your final exam I may offer a problem that requires integration of a function that cannot be carried by a calculator---make sure to work those problems with and without calculator. A couple of problem proposals were poorly written, came with typos, or didn't carry interesting variation of examples (too repetitive). All in all, I would give this section a 3/5.

1. Find 4 different instances of points in polar coordinates that represent the point in Cartesian coordinates given by \( x=3, y=3. \)

2. Convert the point given in polar coordinates by \( (5, 3\pi/2) \) into Cartesian coordinates.

3. Convert the polar equation, \( r = 6 \cos\theta \), into a parametric equation.

4. [I like this problem! It is evil. Good luck getting a solution.] Convert the polar equation, \( 6/r = 5\cos\theta + \tfrac{1}{3}\csc\theta \), into a parametric equation.

5. Convert the following curves (given in polar coordinates) into parametric equations:
   \begin{align} &r^2 = 4 \\ &r\cos\theta = 17 \end{align}

6. Compute the tangent line to the curve given in parametric equations by \( x(t) = \cos^2 t, \) \( y(t) = 2\sin t \) at \( t = \pi/6 \)

7. Find a parametrization for the line segment from \( (0,1) \) to \( (4,0). \)

8. Without using a calculator, sketch the graph of the curve given in parametric equations by \( x(t) = 2t-5, \) \( y(t) = 4t-7, \) \( 1 \leq t \leq 3 \)

9. Express in polar coordinates the curves given in implicit form by \( x^2/16 + y^2/9 = 1 \) and \( 3x-2y=6. \)

10. Find the length of the following curves:
    \begin{align} r &= \theta^2, && 0 \leq \theta \leq \sqrt{5} \\ r &= \cos^3 \tfrac{\theta}{3}, && 0 \leq \theta \leq \pi/4 \\ \end{align}

11. Compute the area of the region bounded by \( r = \sin\theta, 0 \leq \theta \leq 2\pi. \)

12. Compute the area of the region bounded by \( r = 4 + \cos 2\theta, \pi/8 \leq \theta \leq \pi. \)

13. Compute the area of the region bounded by the astroid \( x(t) = \cos^3 t, \) \( y(t) = \sin^3 t, \) \( 0 \leq t \leq 2\pi. \)