OpenLab Assignment 2: Write a question for the first exam

UPDATE 2: LaTeX power users.  Advanced users can check out my additional \LaTeX notes at the bottom of this post.

UPDATE: LaTeX tester.  Want to test out your LaTeX code before you post it in a comment?  There is a LaTeX tester here, where you can type in your formula, hit the button, and see how it looks: http://samples.geekality.net/latex/

NOTE:  When you use the test, do NOT include the dollar signs or the word “latex” — just include the stuff in between.

Assignment (due at midnight on Monday, February 20) .  Create a problem that could appear on the first exam, and post it in a comment as a reply to this assignment.  It should satisfy the following:

  1. It can be from anything we have studied so far this semester, from the first day of class up through Section 6.1, Areas Between Curves.
  2. It must be a reasonable exam problem — not too easy, not crazy hard (I will be the final judge of what counts as a “reasonable exam problem”). Make sure that you can solve your problem.  For ideas, look at the homework assignments, the review sheet, the group work from class, your notes, and so on (you can use these sources as inspiration, but please don’t copy problems from them directly).  If you wish, you can also ask short-answer, explanatory type questions, like  “Explain in your own words ….” or “Why does …” or “What’s the difference between  xxx and yyy?”.  Bonus points for creative questions!
  3. It must contain some kind of mathematical symbols, which must be posted using correct mathematical notation.  How do you do this?  See below.

Extra credit.  Solve one of your classmates’ questions, and post the solution as a reply. Your solution MUST be posted using correct mathematical notation.

What’s the point of this assignment?  Two things:  First, to make you think about what kinds of problems will be on the exam — and creating a problem forces you to consider this from a different perspective (what should be on the exam?) than simply practicing problems.  Second, I want you to start learning how to type mathematics on the OpenLab — how do make integral signs, exponents, square roots, and so on?

Typing math on the OpenLab.  This is not hard — BUT it takes a little getting used to.  Here’s an example. If you type this into a comment:

Here is an integral:  $latex \int x^2 dx$

then (after you post the comment) you should see this:

Here is an integral:  \int x^2 dx

Each equation or expression begins with “$latex” and ends with “$”.  The word ‘latex’, which appears after the first dollar sign, does not refer to the rubbery substance used in hospital gloves and sex toys, but rather to the incredibly powerful and awesome math typesetting language \LaTeX created by computer/math god Donald Knuth (and used by basically all math and science professionals in the universe).  In between “$latex” and  “$” you type your math — many things you type just as they are, like numbers and variables, but each special math symbol has a special code.  In the example above, we use the code for the integral sign, which is “\int”.  To get the exponent on the x^2, use “^” (just like in your graphing calculator).

Here are a few more examples:

Type this: to get this result:
A. $latex \int_0^{\pi} \sin x dx$ \int_0^{\pi} \sin x dx
B. $latex \frac{x+1}{x^2 + 5x}$ \frac{x+1}{x^2 + 5x}
C. $latex \sqrt{x+1} + \sqrt[5]{x+6}$ \sqrt{x+1} + \sqrt[5]{x+6}

Some notes about these examples:

Example A (definite integral): Use curly braces “{ }” for grouping things together. On the integral sign, “_” gives a subscript and “^” gives a superscript, which is how we get the 0 and \pi to appear in the correct places. The code for the \pi symbol is “\pi”. For the sine function we use the code “\sin” (which looks nicer than simply typing in the letters “sin”).

Example B (fractions): The code for fractions is “\frac{ }{ }”, with numerator inside the first set of curly braces { } and the denominator in the second set.

Example C (roots and radicals): Square roots and other roots like these \sqrt{x+1} + \sqrt[5]{x+6} are created using the “\sqrt{}” (for square roots) and “\sqrt[n]{ }” (for nth roots)

Hints and suggestions. Don’t start with a complicated formula. Write a comment with a short bit of math in it, and post it to see what it looks like. You can always edit the comment to make changes.

Stuck? Frustrated? Doesn’t look the way you want it to look? Let me know! Send me an email or simply post a question on the OpenLab — let me know what you’ve tried so far, and what you’re trying to accomplish.

For more examples, this link is a pretty good place to start. Want even more symbols? Here you go.

UPDATE: \LaTeX power user section

You guys have been doing a great job typing math into your comments. You can feel free to ignore this, but if you’re interested in a few additional \LaTeX tricks, read on.

1. Making the vertical bar stretch. For the vertical bar, which we use when solving a definite integral (in the step after we take the antiderivative but before we plug in) to keep track of the limits of integration: just use the vertical bar symbol “|” on your keyboard, usually located above the “\” symbol. BUT if you have a fraction or other piece of math that is taller than a single line, the “|” sign looks awfully small by comparison. How do we make it stretch vertically to match the stuff that comes before? We need to indicate to \LaTeX which part of the mathematics we want it to refer to. We do this by enclosing the pertinent parts in the following keywords “\left.” and “\right|” — note that the first, \left., must include the period after it, and the latter, \right|, ends with a |. Here’s an example

$latex \left. \frac{x^2}{2} \right|_0^\pi $

which renders like so:

\left. \frac{x^2}{2} \right|_0^\pi 

2. That pesky $. If you’re trying to discuss \LaTeX code, instead of actual math, you’ll want to be able to display the stinking dollar sign. BUT the system will automatically compile your code into nice-looking math, making the dollar sign disappear. The solution is to use the HTML code for dollar sign, when you want the dollar sign to appear. The HTML code for $ is $ or, said aloud, “ampersand-numbersign-three-six-semicolon”.

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244 Responses to OpenLab Assignment 2: Write a question for the first exam

    • Jonas Reitz says:

      Hi Lance — you’re off to a good start. Two tips — you don’t need the backslash “\” in front of the number 4, or in front of the curly braces {} — get rid of these and I think your formula should display correctly.

      • lance12 says:

        Professor- I was testing it out. I wasn’t sure how exactly how to delete this post and it just remained there. I’ll make sure to give you a more concrete example. Thank you.

  1. $latex\int_a^{\pi}frac\sqrt{2x+1}{x^10+\sqrt18x}dx$

  2. Your city is about to be the victim of a flood. In order to prepare for it, you decide to build a dam to stem the flood and then reservoirs to hold the excess water.

    To Clarify: There is only one place where the flood can come from– it is one river.

    Because you are so awesome, you happen to know that:
    Water from the flood will flow inwards at a rate of -t^2 + 100t (or any even function with one absolute maximum and no absolute minimum). t stands for time in minutes; the flow rate is a function of time. Assume that all water flow as a result of this function will count of floodwater and will be carried into a reservoir

    How long will the flood last?

    -t^2 +100t = 0 \rightarrow -t^2 = -100t \rightarrow t = 100 , t = 0 \Rightarrow 100 minutes

    Each reservoir can hold 500 ft^3 of water. How many reservoirs are necessary?

    $latex \frac{\int_0^{100} -t^2 +100t}{500 ft^3}

  3. cthoma12 says:

    \int \frac{(\ln x)^2}{x}dx
    u=\ln (x)
    du= \frac{(1)dx}{x}

    \int u^2= \frac{1}{3}(u)^3+du

    Just testing it out !! also if you guys want a template to preview your LaTeX use http://wordpress.com/#!/my-blogs/ i’m pretty sure it’s the same platform as open lab so you can hit preview when u try to make a new post 🙂

    • Jonas Reitz says:

      Thanks, Charles — that’s a nice tip (and yes, the OpenLab is based on wordpress so testing it there is a good strategy). If you don’t want to create a new wordpress site just for testing purposes, you can also check out the LaTeX code tester I posted at the top of the assignment (its http://samples.geekality.net/latex/).

      Your mathematical notation is looking good 🙂

  4. shamonie22 says:

    latex \int_4^{6} \frac{3x^2+\sqrt{x}}{x^2} dx

  5. igorekk132 says:

    \int \frac{x}{\sqrt[5]{x+3}}dx

  6. jishan007 says:

    Jishan Biswas Math 1575 section 6638

    \int x a^x^2 dx

  7. jishan007 says:

    Jishan Biswas , mat 1575 sec 6638

    $latex\int cosx^2 sinx dx$

  8. zbrowne says:

    Zekeba Browne – MAT 1575 Section 6638 M/W 4:00 pm – 5:40 pm

    \int_1^{2}x^2(x^3+1)^10

  9. Joshua Ruiz says:

    Joshua Ruiz – Section 6638

    \int_6^{-1} 3x^2 \sec{(x^3+7)} \tan{(x^3+7)} dx

    • Jonas Reitz says:

      Hi Joshua — nice integral! BUT there is a problem — the function is not defined on the entire interval from -1 to 6 (or 6 to -1). For example, the secant function is undefined whenever cosine equals 0 (since secant = 1/cosine). To save the problem, either 1) remove the bounds and make it an indefinite integral, or 2) change the bounds to a smaller interval on which both secant and tangent are defined.
      Mr. Reitz

    • cthoma12 says:

      $latex \int 3x^2 \sec(x^3+1) \tan(x^3+7) dx \\
      u=x^3+7 \\
      du=3x^2dx \\
      now\ you\ must\ resubtitute\\
      \int \sec(u) \tan(u) du \\
      z=\sec(u) \\
      dz= \tan(u) \sec(u) du \\
      \int z dz =\sec(u) = \sec(x^3+7)+C$

    • Jonas Reitz says:

      Hi Lance — this is a nice-looking integral, but I’m not sure how to find the antiderivative (I can’t figure out how to simplify it, and substitution doesn’t seem to work). Am I missing something? Try working it out — if it’s not possible, see if you can make a change of some kind.
      Mr. Reitz

  10. mmiltz says:

    Melissa Miltz, section #6638 (M&W)

    $\int\frac{2x^{2}+5\sqrt{x}}{x^{-4}}dx+\int\frac{6x^{2}}{5}dx$

  11. mendozak says:

    \int_{3}^{9} f{(x)} - g{(x)} dx

    \int_{9}^{3} f{(x)} - g{(x)} dx

    A. Without any evaluation, explain in your own words which of these problems could yield an area.

    B. In your own words, explain the difference between an area and a definite integral.

    Khoreece Mendoza
    MAT1575 M/W 4PM

  12. Find the area that is formed by the curve of y = sin x and the x-axis from {0 , π}?

    Find the area that is formed by the curve of y = cos x and the x-axis from {-π/2 , π/2} ?

    I think these are reasonable problems and believe that if you can solve these, you understand the concepts and i think you’ll be able to do good on the exam?

  13. Jonas Reitz says:

    Hi Gurpreet — I like these problems a lot! (but I notice you managed to avoid using \LaTeX — post something else or repost with LaTeX for full credit).

    Regarding your final question/comment, I’m not sure whether you are referring to a) your two problems above, or b) all the problems appearing in the comments so far. Certainly if you can do all the problems appearing so far, you’ll be well on your way to an A on the exam (but the examples we have are not exhaustive — there will be other things on the exam as well!).

    Mr. Reitz

  14. bettygeorge says:

    bettygeorge MA1575,6638

    $latex\int_1^3(1/t^2-1/^4)dt$

  15. vasquez says:

    disregard \int_1^1(1+2y)^2 dy… i made a mistake writting it….

  16. vasquez says:

    $latex\int_1^2(1+2y)^2 dy$

  17. ymerej613 says:

    Jeremy Li
    Mat 1575 6637
    \int\ \sqrt{5x^2+10} dx

  18. koshygkoshy says:

    Bibin Koshy
    Mat 1575 6637
    $latex\int_2^{4}\frac{dx}{x^2 + 5x}dx$

    • Jonas Reitz says:

      koshygkoshy — the notation looks fine (if you insert a “dx”), but the integral is hard! I’m not sure how to find the antiderivative — see if you can work it out, and either post a solution or modify the problem. Thanks,
      Mr. Reitz

  19. bbravo999 says:

    Evaluate the indefinite intergal:
    $latex\int \frac{cos\sqrt{x}}{\sqrt{x}} dx$

  20. bbravo999 says:

    $latex\int \frac{cos\sqrt{x}}{\sqrt{x}} dx $

  21. bbravo999 says:

    latex \int \ frac{ cos\ sqrt{x}}{\ sqrt{x}} dx

  22. bbravo999 says:

    Evaluate the indefinite intergal:

    $latex\int \frac{cos\sqrt{x}}{\sqrt{x}} dx $

  23. bbravo999 says:

    Evaluate the indefinite intergal:
    \int \frac{cos\sqrt{x}}{\sqrt{x}} dx

  24. takther2009 says:

    $latex/int_2^{/3}/frac{dx}{(x-2)}$

  25. Keyla says:

    latex$ \int_2^{4x^5}\frac{\sqrt{\sin x+1}}{x^2}dx$

  26. tonymei999 says:

    Jiarong Mei section 6638

    \e^{x}*sqrt{1+{e^x}} dx

  27. theozeng says:

    Theo Zeng section 6638

    \int sec^4 x tan x dx

  28. debitcard says:

    sub
    u= $latex\{e^x+1}$
    du= $latex\{e^x dx}$

    = $latex\int\sqrt{u}du$

    $latex\int\sqrt{u}$ = $latex\frac{2u^3/2}{3}+C$

    substitute
    $latex\{e^x+1}$
    for u

    =$latex\frac{2/3(e^x+1)^3/2}/+C$

  29. jishan007 says:

    Jishan mat1575, section 6638
    \int_1^4\ln x dx

  30. justinblaize says:

    $latex \int lnx+4x^2-2x+4

  31. debitcard says:

    Koonhoi Xie MAT 1575 – 6638.

    Find the integral between two curves given the function x & x^2

    Hint*- Start by finding the point of intersection.

    Formula is given by $latex\int_a^bf(x)-g(x)dx$

  32. kedeshia1111 says:

    \int\frac{\sin {x}}{1+\sin{x}}dx

    • Jonas Reitz says:

      Hi Kedeshia — you just need to put “$latex” at the start and “$” at the end, and your code should display correctly. However, I think your problem is hard — I can’t figure it out! Try it, and see if you can make it work, or make a modification.

  33. nazir ahmed math1575-6638

    \int x^3lnx dx

  34. KHonda says:

    $latex\frac{x}{\sqrt[6]{x+3}$

  35. Antonio Downer
    Math 1575-6638

    \int\frac{\sqrt{4x^2+1}}{8x} dx

  36. danytrueman says:

    $late\int(1-t)(2+t^2)dt$

  37. $latex\int\ 2x(x^2+4)^{100}$

    • Jonas Reitz says:

      Hi Tiff — you’re close! Get rid of the backslash in front of the x, and it should display properly. Quick question: is this a “find the derivative” problem? If so, convert the x’s (except the first one) to t’s, and add the instructions “find the derivative”.

  38. alamin163 says:

    Find the area between y=x^2 and y=x^3 between x=0 and x=1 when the function is given:
    &latex \int (x^2 – x^3) dx$

  39. A company produces \int_\frac{x^2}{\sqrt{X}} solar panels an hour. How many panels will be made in six hours?

    • Jonas Reitz says:

      You’re off to a good start, but the question needs a bit of modification — I’m not sure how to solve it from the information given! For notation, don’t forget to include the “$latex” at the start, and “$” at the end.

  40. Drake says:

    Drake Li Section 6638

    \int_0^{\pi} \csc x + sec x dx

  41. darklor7 says:

    Alexander Barbaran 7216

    \int_0^{3\frac\pi} \frac \sqr{x+3} \sqr{x-6}

    • darklor7 says:

      it doesn’t work if i use \int 0^{3\pi} \frac \sqr{x+3} \sqr[3]{x-6}

      • Jonas Reitz says:

        Hi darklor7 – you’re on the right track! You need to put an underscore “_” before the zero to make it show up as the lower bound of integration. The code for the radical sign is \sqrt, not \sqr (don’t forget the t). Finally, after \frac, the numerator and denominator should each be enclosed in curly brackets { }, so it would start like this: “\frac{\sqrt{x+3} …” and so on. Give this a try and see if you can make it work.
        -Mr. Reitz

  42. jesus22 says:

    a problem that I always didn’t know how to solve but one we took the test i lean how to solve it

    $latex\int_0^{5}{\frac{\sqrt[3]{27}}{\sqrt[3]{{(x-3)}^4}}}dx$

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