ORGANIZE YOUR RESOURCES (CALCULUS)

Since this is my first year focusing on the Calculus curriculum, final exam time is the perfect opportunity for me to refresh my own study skills. For those who struggle with preparation for the cumulative exam each semester, perhaps seeing how someone else studies could give you some ideas of your own.

My situation is slightly different than the students’. I have 4 Calculus students in 3 different schools, 4 different teachers, and 3 different textbooks (one of which is a newer version of another with many of the same problems, some new ones, and all on different pages!). The organization needed to remember where each student was in the various schedules and what homework I had completed in preparation for tutorials for whom got the better of me and was in need of serious revamping even before I began over Winter Break to study for finals. At that time, I collected all homework and piled up assignments by student, then within each student’s pile, I put things in chronological order.

Some teachers had flitted around the text, while others had followed the order given in the book, so when one student was working on Inverse Trig differentiation, another was working on Related Rates. My solution for organizing was to make 4 lines of work (one for each student), assembled by TOPIC. And that gave me the idea to make my own workbook with notes relating to topic and separated with tabs so I could access my backup materials as needed.

MY TIP TO YOU
You already know I expect you to keep all class notes, homework assignments, tests and quizzes (from the first day of the semester until the day you retire to a nursing home). If you’ve been following my suggestions, this material is already organized by chapter or unit according to the chronological order established by your teacher.

That’s a lot of paper and probably difficult to use as reference. I suggest you acquire either the “Stickies index tabs” (mine are the Staples brand but I think 3M has a version also) or actual binder tab sheets. Mark the tabs according to the topics you’ve studied and separate your work by topic. When you come across a final review question that relates to the Mean Value Theorem, for example, you can easily find your past work on that subject.

Here’s a list of the topics in my newly organized, personal reference binder:

FUNCTIONS
LIMITS

RULES OF DIFFERENTIATION (this is my universal sheet with all of the examples including derivatives of Trig functions, e, and all that sum, difference, quotient, chain rule stuff in one location. Not every text provides the handy list on the book cover, so I have my own to use as reference if I forget a rule or just want to verify my work.)

VELOCITY/ACCELERATION
EXTREMA
MEAN VALUE THEOREM
F’’
OPTIMIZATION
NEWTON’S METHOD
IMPLICIT DIFFERENTIATION
RELATED RATES (by far the most voluminous section complete with examples from the internet. I defy any teacher to find a problem I have not researched and copied the algorithm for solution.)

Finals are fast approaching. You may already have your Final Review Packet. I’m thankful to have the organizing completed before trying to work on 4 entirely different sets of problems from 4 uniquely disparate teachers!! Your study plan will surely be less complicated than mine, so take heart in knowing that STUDYING SMARTER, NOT JUST HARDER has many positive rewards.

THE AMBIGUOUS CASE (Geometry and Trig)

Remember triangle congruence in Geometry? In what ways can you prove two triangles congruent? SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), HL (hypotenuse-leg).

But why can’t you use SSA (side-side-angle)?

BECAUSE IT’S THE AMBIGUOUS CASE! (What a great SAT word - ambiguous - meaning “unclear”)

Let’s take 2 sides of a triangle and the non-included angle (SSA).
We know the lengths of the two green legs and angle A, so the dotted line is the path that line AC will take even though we don’t know how long it will be.

But we don’t know angle B, so line BC can swing either left
or right

Notice that angle B could be either acute or obtuse. When you use the Law of Sines to find angle B, the calculator only gives you an acute angle. This will certainly be one value of angle B, OR THE ANGLE YOU CALCULATE MIGHT BE THE REFERENCE ANGLE OF OBTUSE ANGLE B!

How do you know if the obtuse angle is really possible ? This is the AMBIGUOUS part. You have to “think” about it, or rather, you need to "calculate" about it.

Take the angle you know (angle A) and add the angle B that was the result of the Law of Sines. By subtracting the sum from 180°, you will have one value of angle C.

Now use the angle calculated through the Law of Sines as the reference angle to find the other possible value of angle B. Add known angle A to it. Are there any degrees left for angle C? (Is the sum still less than 180°?) If angle A plus obtuse angle B is less than 180 degrees, you have a second possible triangle from the given information.



If you’re a fan of math, the rigorous explanation might appeal to you:

Given 2 sides and the non-included acute angle and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), this is the ambiguous case and two different triangles can be constructed.

Given: AB
Given: BC
Given: angle A, where angle A < 90°

If BC > AB(sin A)
and BC < AB,
the you have an AMBIGUOUS CASE.


Here are a couple of questions to add depth to the concepts relating to the Ambiguous Case:
1. The triangle congruence rule AAS (angle-angle-side) is really a special application of another triangle congruence rule. Which one is it? Explain why. (Hint: Think of the "no choice" rule.)
2. Why does the calculator only give you one possible value of angle B? (Hint: Think about the quadrants of inverse sine, inverse cosine, and inverse tangent.)
3. What unique characteristic of the unit circle (and inverse sine) applies when you use acute angle B as a reference angle? (Hint: Think about the sum of the degrees in a triangle and the degrees in a straight line or semicircle.)
4. The rigorous explanation emphasizes that BC must be shorter than AB. Why? (Hint: Look at the diagrams of BC swinging left and right. What would happen if BC was longer than AB?)
5. Why does the rigorous explanation stress that the known angle must be less than 90°? Why couldn't it be 90°? (Hint: Think about HL.) or obtuse? (Hint: Think about questions 1, 2, 3, and reference angles.)

When you think about it, there really isn't anything new in your math study......just more ways of using what you already know!!