Chloe Urbanski Wawrzyniak

Recommended Resourses

• Precalculus Practice Problems:
  • http://www.teaching.martahidegkuti.com/shared/lnotes/5_trig/equations/equations2/equations2.pdf
  • https://www.ixl.com/math/precalculus/solve-trigonometric-equations
  • https://www.khanacademy.org/math/trigonometry/trig-equations-and-identities/advanced-sinusoidal-equations/e/solve-advanced-sinusoidal-equations
• Khan Academy Exam Review: If you are looking for a resource to help guide you through studying for the final exam, I think Khan Academy is a good one. Below, I've listed links to the home pages for relevant topics to this course. Once you get to the homepage, you have a few options, depending on how you study best. The "Explore" tab gives an overview of the resources available there. The "Practice" tab lists topics that you can choose from, and then it will give you practice problems on that topic. This is a good option if you're looking for help on a specific topic. If you are looking for some overview review and help figuring out what topics you need the most help on, I recommend the "Mission" tab. It should automatically bring up a diagnostic pretest (if not, it's listed at the top as the "Mission Warm-up". That will guide you through sample problems to determine which topics you've mastered and which topics you need more practice on. Once you finish the warm-up, you can practice problems that it indicates you need help with. The links for the homepages are listed below:
  • Trigonometry
  • Calculus
  Please don't forget to first create an account and register with me as your instructor. The directions for this are listed below.
• Cubic Equations: In workshop 9, we worked a problem which was considered by the mathematician Niccolo Tartaglia. He is probably best known for his role in the history of solving the cubic equation (ie finding something like the quadratic equation but for order 3 polynomials instead of order 2). I find this story particularly interesting, so I wanted to share it with you. This wikipedia page talks all about the solution that was eventually discovered. The pertinant section is in history, starting with "In the early 16th century...".
• Riemann Sums: It's easy to get caught up with the notation in these problems, so I've found a couple of resources to help you with these computationally messy problems:
  • Wikipedia: This page has more notation than you need to worry about, but it does have some helpful ideas. It lists a couple different types of Riemann sums (left, right middle) as well as an example for each. If nothing else, this page has some great pictures and animations that show the picture of what's happening with Riemann sums and how they relate to integrals (which are the area under a curve).
  • Wolfram Mathworld: This page has less explanation of what's going on, but it has a great applet that lets you see what different Riemann sums would look like for various functions. In other words, it shows the graphs of the functions and the rectangles for the Riemann sum. I recommend playing with the applet to get a picture in your mind of what's happening. Try different functions and then try different types of sums with different Δx values.
  • Harvey Mudd Notes: This page has a bunch of good pictures that show a progression from the actual area under a curve to the Riemann sum. I should note that it does the problem in more generality than we've discussed so far: it doesn't assume each subinterval has the same length (it gives each one its own length), and it doesn't require our height to come from one of the end points or the midpoint (it allows any point in the subinterval to determine the height of that rectangle). This is more general than you need to be able to do computation for, but the pictures are great for getting an idea of what's going on.
• Optimizaiton: Students tend to have difficulty with optimization problems. I've found a number of online resources to check out if you need the extra help:
  Wikipedia
  • Khan Academy
  • Drexel University Lecture Notes
  • UC Davis Lecture Notes
• Sorting Algorithms: Workshop 8 had a problem which dealt with running times of various sorting algorithms. This is a youtube video which demonstrates a number of sorting algorithms out there. If you are a CS major, I think you will find it interesting.
• Lots of Practice Problems: UC Davis has this awesome bank of practice problems for calculus students. They don't have every subject, and I've only glanced at a few pages, but it seems like a fantastic resource. As you study for exams, you should consider using this webpage for practice in the topics you are having trouble with.
• Related Rates: I've typed up some notes on related rates. Students tend to struggle with related rates, I wanted to provide some extra resources for you. This version of the notes was uploaded on 10-31-2016. Corrections will be made if errors are found.
• Squeeze Theorem: Here is the wikipedia page on the Squeeze Theorem. It gives a good description of the theorem, both technical and intuitive, as well as a proof of the theorem (not something you need to know for this class, but if you are curious, then you do have all of the tools you need in order to read and understand it) and a good variety of examples.
• Mathematical Symbols: Here is a quick reference for nearly any mathematical symbol you might encounter in this class.
• Delta-Epsilon Proofs: If you are having trouble understanding how to write delta-epsilon proofs, you can check out this, this, or this . You'll notice some notation in the last two links that you may not be familiar with. See here for a list of mathematical symbols. You'll find the ones from those resources listed under "Symbols that Point Left or Right" and "Symbols Based on Latin Letters".
• Intermediate Value Theorem: I really like this page for explaining Intermediate Value Theorem. It goes throught the intuition, has lots of pictures, and even discusses why it's so important and useful through some interesting examples.
• Intuition for Formal Definition of Limits: Here is a webpage that I found which gives a thorough explanation of the intuition behind limits and their formal defintion with epsilons and deltas. If you are having trouble getting a feel for limits, this page should help you see the picture of what's going on.
• Limits and Infinity: When we add infinity into our concept of limits - whether we are taking limits at infinity or our limit evaluates to infinity - our formal definition has to change a bit. This page is a bit technical, so it's not so great for first intuition if you don't feel comfortable with limits yet (see previous resource), but it does go through how our definition has to change, why it has to change, and a few examples using the changed definition.
• Inverse Trig Functions: In recent sections, we've been using inverse trig functions in our examples. You need to be comfortable with these functions. This Khan Academy page has a number of videos discussing them, and you should absolutely look at this if you are not yet totally comfortable with them.
• Kahn Academy: Khan Academy is a free website with a number of instructional tools, including pracice problems and instructional videos. If you would like to use their resourses for this class, please folow the following set of instructions:
  1. Go to khanacademy.org and sign up for a free account, or log in if you already have one.
  2. Go to khanacademy.org/coaches
  3. When prompted, enter the course code: NHTSZ6. This will add you to the course, with myself as an instructor.
• IXL IXL is essentially a bank of interactive practice problems. Unfortunately, membership is not free. However, you can do as many practice problems as you like without a membership account. You just cannot save your work or track your progress. You'll notice that the design of the website seems a bit jouvenile for a college course. This is because the majority of their resources are for below college level. However, they have a section on calculus as well as a section on precalculus if you need to brush up on some old skills as well.
• Math Tutoring At Rutgers: There are a couple of tutoring options on campus which are free to you as students. The math department has a webpage with a couple of links to these resourses. I've not looked much into them yet, but I will add more here as I learn more.
• Precalculus Diagnostic Test: This is the test that you took on the first day of workshop. You should work back through those problems and make sure you are able to do them all comfortably. If you were not in the first day of workshop for any reason, let me know in class and I'll give you a blank copy. Here is a PDF of the answers. Here is a blank copy for you to practice from.

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