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!!