MATH 141 Fall 2014 Review Exam (5|5)

Sections 11 and 12

Nice selection of problems, with balanced difficulty from very easy to boss level. I am missing some nice problems of sequences — remember that there will be some of those in the final exam! The problems with series are cool, though.
Some of the integrals are extremely tricky, and as a matter of fact, there is one that I don't believe anybody can get — unless you use some techniques of integration that we have not learned in this first approach to Calculus.
I decided to let that one evil integral in the exam, to torture all y'all.
Oh wait! You guys know about wolframalpha... Nevermind
  1. Compute the derivative of the following functions:
    \begin{align} k(x) &= \int_{3x}^{9x} \frac{z^2-7}{z^2+7}\, dz \\ g(x) &= \int_0^x \sqrt{1+t^2}\, dt \\ f(x) &= \int_0^{x^2} \sin t\, dt \end{align}
  2. Compute the following limit:
    \begin{equation} \lim_{n \to \infty} \sum_{i=1}^n \frac{6}{n} \bigg[ \big(\frac{2i}{n} \big)^2 + 5 \big(\frac{2i}{n}\big) \bigg] \end{equation}
  3. Employ the Fundamental Theorem of Calculus to evaluate the following definite integrals:
    \begin{align} \int_1^2 \frac{4+u^2}{u^3}\, du &&& \int_6^8 x^4 + 20x^2 +10\, dt \end{align}
  4. Compute the following indefinite integrals:
    \begin{align} &\int \sin x + 2\tan x\, dx && \int \sec^2 x + \cos x - \cot x\, dx \\ &\int \big( x-\tfrac{1}{x} \big)\, dx && \int e^{5x+8}\, dx \\ &\int \frac{1}{x\ln x}\, dx && \int \frac{2x^6 + 8}{(7x^7 + 8x)^3}\, dx \end{align}
  5. Find the net area of the following functions, in the indicated intervals:
    \begin{align} f(x) &= \lvert x \rvert && [-1, 3] \\ f(x) &= \sin x && [0, 2\pi] \end{align}
  6. Find the sum of the following series:
    \begin{align} \sum_{n=1}^{300} 4n^3-6n+18 &&& \sum_{n=1}^{3000} \frac{n^3-4n^2+3n-7}{15} \end{align}

Sections 15 and 16

This is a more complete and thorough review exam for this part of the course. I am still missing some nice problems with sequences — we only have one, from Tierra. I am missing maybe a net area problem, but that is not so terrible, because you posted a decent selection of definite integrals.
Except for those missing problems, I have to say that this exam is very close to the level of difficulty and volume of questions that you are to expect in my final. Good job!
And there was also two students in this section that came up with really evil integral. One of them is doable with substitution, but it takes a trick we haven't learned yet — you will see it in MA142, though. See if you spot it! The other two... not so much. They were so extremely hard, that I had to take them out of this review test.
  1. Find the next three terms, and the general term of the sequence that starts as \( -\frac{1}{2}, \frac{1}{4}, -\frac{1}{6}, \frac{1}{8}, \dotsc \).

  2. Evaluate the following limit:
    \begin{equation} \lim_{n\to \infty} \sum_{i=1}^n \bigg[ \big(\frac{i}{n} \big)^2 + \frac{i}{n} \bigg] \end{equation}
  3. Compute the following sums:
    \begin{align} \sum_{k=1}^{200} 7k^3 - 3k^2 + 15 &&& \sum_{n=1}^{1000} (4n+n)(2n^2+7) \end{align}
  4. Compute the following integrals:
    \begin{align} &\int 3x^3+2x+5+\frac{6}{x^2}\, dx && \int \frac{1}{4}x^2+\frac{2}{3}x+\frac{19}{7}\, dx \\ &\int e^x \sin( e^x) \, dx && \int \frac{( \ln x )^2}{x} \, dx \\ &\int_1^e \frac{1}{4x}\, dx && \int_1^2 \frac{e^{1/x}}{x^2}\, dx \\ &\int \sqrt{18x} + 89\sin x\, dx && \int \sin^3 x\, dx \\ &\int_1^{2e} \frac{5\ln x}{3x}\, dx && \int \frac{186\cos x}{34 \sin x}\, dx \\ &\int (6x-4)(3x^2-4x)^3\, dx && \int\frac{523x^2}{(4x^3-7)^{1/5}}\, dx \\ &\int \frac{24x^3-15x}{2x^4-5x}\, dx \end{align}
  5. Use the Fundamental Theorem of Calculus to find the derivative of the following functions:
    \begin{align} f(x) &= \int_{x^3+4x+\ln x}^{2x^2+4x+e^x} \sin t\, dt \\ f(x) &= \int_{5001}^{\ln x + 4^x-17} e^x+10\, dt \\ g(x) &= \int_x^3 \sin\big(\sqrt{6t}\big)\, dt \\ g(x) &= \int_{e^x}^2 3\sin^8 t\, dt \end{align}