AP CALCULUS AB

This information was a gift to me from Calc Student Varun. Many years of retired AP tests WITH ANSWER KEYS. Not just Calculus, either; find other subjects in the left column by selecting "courses/tests."

http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html

SPECIAL STUDY FOR THE AP CALCULUS AB TEST

I’m starting study for the AP Calc AB test, a mere 4 weeks away. I’m using the Barron’s text 10th Edition and completing the practice exercise questions in each chapter. My students are following suit, working independently through the Barron’s test prep book AND, of course, keeping up with the classroom assignments as directed by their teachers.

Here’s the plan.
1. Take the Diagnostic Test. I was shocked at how much I had to review for some of the basic questions from the beginning of first semester. The Diagnostic served as a motivator to get me moving and gave me a rough outline of what to study. I did this over Spring Break so there would be time to follow through on the study that was indicated.

2. I don’t find the “lessons” to be very helpful. But when I run into trouble, I’m reviewing from both the classroom text and the Barron’s descriptions.

3. Complete the Practice Exercises, a few at a time. There are gobs and I DO have other things on my to-do list. Sometimes I may work 3 or 4 problems; other times I have an hour or so and can work through more at once. But at the end of each study time, I check my answers and highlight the questions that I missed. The next study period, I review the missed problems before starting any new ones.

4. HERE’S THE IMPORTANT STUDY STEP: I have a separate piece of paper next to me when I’m correcting my work. I like card stock because it’s easier to keep track of and there will be a LOT of paper generated during this study program. When I run into a concept that caused the loss of points, I make a note of it on the “study guide.”

I’m actually creating a composite of all the information I need to review again and again until I can employ it at will. Beside the concept, I indicate the page and problem number that I will work again after the Practice Exercises for that chapter are complete. I’ll keep reworking these problems until I can do them perfectly (and quickly).


Why do I study so much before the AP test? There are several ways that the AP score might be used. Some teachers make the score part of the classroom grade. Some students (myself included) are personally committed to... (how can I say this without sounding dangerously like a nerd?)...well, there’s no way to get around it....committed to the highest possible score. And we obsessive test takers are not alone: some universities require a “perfect score” in order to count the grueling high school work as college credit. Earning an A in high school is nice; getting a good score on a high ability test is great; but earning college credit is why many students take AP courses in the first place, so it’s worth the effort to save the money and time required to retake the same course in college.

So, let’s get started.....differentiation, page 139, question 34 out of 101 in this section. If you have unanswerable questions or suggestions for solutions or study strategies, email tutoring.resources@yahoo.com and indicate "Calculus 911."

COMPLEX NUMBERS...TO POLAR COORDINATES...TO RETANGULAR COORDINATES

What’s a COMPLEX NUMBER? We know about Real Numbers, Imaginary Numbers, and Undefined Numbers, but these are all relatively simple and stand alone. What if we combine a Real Number and an Imaginary one through addition or subtraction? (Sometimes using the Quadratic Formula gives this type of answer.) It’s a little more complicated and so “complex” is a good descriptor.

To graph the complex number on a coordinate plane, the X-axis represents the Real Number and the Y-axis represents the coefficient of the Imaginary Number.

There’s another way to measure where point (3 + 4i) is on the coordinate graph. What if we measured the angle around the unit circle and then how long the ray is from the origin? This would give us POLAR COORDINATES.


Oh, look!! A right triangle!! ( Thank you, Ms. Smith, your 9th grade geometry teacher who made sure everyone knew the Pythagorean Theorem very well! Not to mention SohCahToa!)

To find the angle between the ray and the x-axis, we could use Tangent.

The opposite side of the triangle is the coefficient on the imaginary number (b) and the adjacent side is the Real Number component (a) of the Complex Number.

To find the length of the ray, we use the Pythagorean Theorem:

where a and b come from the complex number and c is the length of the ray.

From our complex number example, a = 3, b = 4, and c = 5, while θ = 53.13 degrees or .93 radians.


So, we could locate the starred point on the graph by using POLAR COORDINATES.

In polar coordinates, we aren’t looking for the intersection of X=some number and Y= some number, but where is the angle located on the Unit Circle and how long is the ray extending from the origin. The ordered pair is

Read this ordered pair..."are, theda".......where have you heard this “phrase” before? Remember angular velocity? The length of the arc on a circle is s=rθ. [If you are graphing Polar Coordinates, recognize a NEGATIVE 'r' starts at the origin but moves in the OPPOSITE DIRECTION and a NEGATIVE theda moves in a clockwise direction. Can you think of 4 different ordered pairs to indicate (4,π) on the Polar Graph using negatives?]

We could also describe the location of the star by using TRIGONOMETRIC COORDINATES.

For this transformation, we should remember from Trigonometry (or preview if you haven’t taken Trig yet) that the y-axis represents Sine and the x-axis represents Cosine.

These are, in fact, the equations you would use to convert from Polar Coordinates to Rectangular Coordinates.


Here’s a typical problem:

Convert the complex number to Polar Coordinates and Rectangular Coordinates.



[Print out this reference box and add it to your review card.]

THINKING QUESTIONS:
1. How would subtracting the imaginary number effect the placement of a point on the graph?

2. How would this change the Polar and Rectangular Coordinates?

3. How do you use inverse trig functions and reference angles to find angles not in quadrant I?

Doesn’t this stuff make you think in CIRCLES? Or maybe it just makes your mind go round and round. GOOD! Because next we’ll be looking at Cardioids and Limacons (with and without interior loops). If you want to see a really impressive, interactive display of the new concept, go to

http://www.intmath.com/plane-analytic-geometry/ans-8.php?a=1

and be amazed!!

MATH 911

For our Wisconsin students and their friends. During this time of potential, unexpected school closings, keep up with your academic growth. Continue moving through your textbooks at the pace that would be normal in your classroom: read, answer discussion questions in writing, complete textbook reviews, and take chapter tests. If you run into difficulty with any of the Math material, contact Tutoring Resources through email for assistance:

tutoring.resources@yahoo.com

and type "Math 911" in the reference box. One of our tutors will get back to you ASAP.

TAKING THE SAT?

The next SAT test is March 12. Are you preparing for it? It’s time to get started if you haven’t already.

Purchase a study manual: one that contains REAL SATs from the publishers themselves. Do NOT rely on another publisher to create an SAT-like test. Use the real McCoy so observations that you make during study will have direct application to the actual test.

1. Take a test and score it so you know where the points come from.
2. Observe the format (which is parallel to the PSAT that Juniors took in October, but is very different from the ACT that Juniors will take in April).
3. Check your pacing.
4. Determine your strategy for omissions.

Here’s an SAT tip for the Reading section. Notice where in the text you find answers to each question. Notice anything? If not, underline the statements within the text that support the correct answer. Number this highlighting with the question number. Notice it now?

As a general rule, the questions are asked in the order the answers appear in the passage. Read the first question, then read the text until you come to the answer. In most cases, you can go on to the next question and continue reading until you come to that answer. If you get to the answer for a subsequent question, you know you’ve missed the important clues. Line references help here, but in some cases you may need to know what the next question is also.

With this strategy, you are reading the entire passage while collecting points. No wasted time and a direct link to where discrete answers can be found.

Be prepared for the essays to be bone dry. Just resolve to find correct answers and don’t worry about the lack of interesting topics.

Remember to “study smarter, not just harder.”

LINEAR AND ANGULAR VELOCITY

Trigonometry students have this to look forward to, but those in College Algebra may be reexamining these issues right now. Although linear and angular velocity questions can be answered using Geometry concepts alone, calculating circumference and using degrees can create long solutions with lots of opportunities to make silly, little mistakes. Here are the equations you need in order to turn these problems into simple substitution work.

First, take a look at the names of these concepts: LINEAR deals with lines and is the familiar “miles per hour” measurement. In circular motion, it's the line measure we call circumference. ANGULAR deals with angles in a circle -- in common terms RPM, or revolutions per minute. Velocity, as we commonly know it, is distance divided by time.


To make things simple, we can calculate Angular Velocity first and use it to quickly find Linear Velocity.

Remember in Geometry that we calculated the arc length.


Given a radius of 4 and a central angle of 50... S = (50/360)• 4•4π = 20π/9.


From Trig, we can convert the degrees to radians:




That’s your first equation...


To find Angular Velocity, we want the central angle measured in radians, θ, divided by time. Oh, and we get a new, cute symbol that some will be calling “W,” but what is actually the Greek letter “Omega,” ω.

That’s your second equation...
If the question gives you RPM (revolutions per minute), you can convert rpm to radians by multiplying by 2π, the radians in each time around the circle. Don't forget to convert minutes if a different time measurement is requested.

So starting with the old velocity equation, substituting S (arc length) for distance, then substituting r θ for S, and finally substituting ω (omega) for θ over t , we have the final equation: Linear Velocity.


If equations are difficult for you to remember, try printing up this visual display:


Linear velocity is the outside equation; angular velocity is the inside equation.

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.