Things I expect you will be able to explain on the final exam

The Goal Please know that math is isn’t a list of 1,000 things to remember. The idea is really to teach you how to think, albeit quantitatively. As a result, I will ask you to explain concepts or analyze critically and verbally. In general... 90% of math can be done without a calculator. Formulas are nothing to memorize (after all, you can always look them up in the real world). Rather, if you understand where they come from and maybe understand how they look (graphically), then you will understand with a depth that supersedes memory. Don’t give up. You know the answer is there, and you know that I will not ask you to solve something for which you don’t yet have the tools.

What follows is a list of things that I expect you to understand and be able to explain by now. All of this is fair game for the exam, and if you don’t understand any of these things, remember that we have videos made for each of these items. Also, you are welcome to come see me for extra help: Monday or Tuesday after school or any day before school.

The Material (Right Triangles) What are the six trigonometric definitions? You should know these by heart by now both in terms of SOHCAHTOA and in terms of their x,y,r ratio equivalences. How can you use special right triangles to know the sine, cosine, tangent, etc. of angles without using a calculator? And where do those ratios come from? How do you use inverse functions to determine angles of triangles if you know the sides?

The Material (Oblique Triangles) What is the Law of Sines? How is it derived, and how and when is it used? When do you know when there is more than one possible triangle or not (the Ambiguous Case). Then, how do you deal with finding more than one possible triangle? What is the Law of Cosines? How do you use it to solve oblique triangles? Why would you use that instead of the Law of Sines?

The Material (The Coordinate Plane) Where is each trig function positive and negative and why? What are angles in standard position (positive and negative)? Coterminal angles, and reference angles, and how do you find them? What are radians, and how do you convert from angles to radians and vice versa? And why would you do it that way? How do you find the sine, cosine, tangent, etc. of different angle measures from 0 – 360 and different radian measures from 0 – 2pi? For example, you should be able, without a calculator, to find the exact value of tan(7pi/6) and explain how.

The Material (Trig as a Function) Are sine, cosine, tangent, etc. functions? How do you know? What is the domain of each function and why? What is the range of each function and why? What are the graphs of each function and why? What is the period of each function and why? How can those graphs change? What are the different elements of a trig function and how do those elements change the graph of a function? (amplitude, frequency, period, phase shift, vertical translation). How can you determine the function from a graph?

The Material (Analysis) What are the negative angle identities, and why are they true? What are the reciprocal identities, and why are they true? What are the quotient identities, and why are they true? What are the cofunction identities, and why are they true? What are the periodic identities, and why are they true? What are the pythagorean identities, and why are they true? How can all of the identities be used together to simplify trigonometric expressions? (In other words, you should be able to use all of the identities together to verify that one trigonometric expression equals another).

The Material (Advanced Analysis) I will not ask you to prove the angle addition or subtraction formulas. I will not ask you to prove the double or half angle formulas. I will not ask you to prove the power reduction formulas. But you may have to use these in the process of some trig verifications. Also, you should know how to solve trig equations. This involves the use of inverse functions and the proper application of 2πk and πk.

The Material (Sums & Series) You should know and be able to explain the difference between an explicit formula for a series and a recursive formula for a series. You should know and be able to explain the difference between a geometric and an arithmetic series. You should be able to tell why the sum of an infinite geometric series is not findable when the absolute value of r is greater than 1.

The Material (Limits & Calculus) You should be able to explain what a limit is. You should be able to give some examples of when you can’t find a limit for a particular function. You should be able to explain either of the formulas for the derivative. You should be able to tell what a derivative is... What it means.

The Grading Each problem will be worth 10 points with my normal criteria for partial credit.

Tips Make sure you can explain all of the above. That would be a good project for this weekend. Make sure you can do all of the problems in the review packet I will give you on Monday. If you are having trouble remembering a topic, look for the video on it. Prepare one sheet of notes (front and back = OK) for use on the exam. Manage your time. When you get the exam, look through it and give yourself a budget for how long to work on each problem.

The Truth I take great pride in the fact that I will not lie to you. If you are good at something, I will tell you. If you are being stupid, I will tell you. So, believe me when I say this: This exam will be very difficult. It isn’t that I’m trying to make it difficult, but it will be comprehensive. We have covered a ton of material this semester. So, the exam, by its very nature, will involve both breadth and depth. I want each of you to do well. No lie. But that is going to take some serious preparation on your part.