Tuesday, January 30, 2007

2000 Calculus AB Free-Response: Question 1 Part B

Lets continue now with Question 1 Part B.

Part be asks that we revolve f(x) and g(x) around the x-axis. To do this, must integrate f(x)^2 - g(x)^2 using the same parameters as Part A (0 to .94194). Then, after it is integrated, we must multiply it by pi.

Remember that the first part of the AP Calculus free response allows you to use a graphing calculator. A graphing calculator is actually required to solve this problem because the indefinite integral of e^(-x^2) can not be found.

Back to the problem... just put in the definite integral in your calculator and press enter. If you have a TI-83 Plus or TI-84 Plus, type fnInt((e^((-2x)^2)-((1-cos(x))^2),x,0,.94194).

You should get Volume = 1.7466 units^3

Friday, January 19, 2007


Today, while sitting in my physics class, I saw a very funny shirt that one of my peers was wearing. You need some knowledge of physics and calculus to understand this one. Take a look below.

Still don't get it? You need to know that s is displacement. The derivative of s with respect to time, (d^2)s/(dt^2), is acceleration. The derivative of acceleration, (d^3)s/(dt^3), is called jerk. So the shirt read "DON'T BE A JERK."

Pretty funny, right?

Thursday, January 4, 2007

2000 Calculus AB Free-Response: Question 1 Part A

In my last post, Learn from the past AP Tests: 2000, I linked to the 2000 AP Calculus AB Free-Response questions. Today, I will start to go through the solutions, starting with question number one, part A.

Question number one, part A asks for the area of a region, R, between two functions (also known as curves), which I will call f(x) and g(x). f(x)=e^(-(x^2)) and g(x)=1 - cos(x)

To solve, we must first find the area under the top curve, f(x). To do this, we must integrate f(x) (Also known as taking the antiderivative. The antiderivative of f(x) can be written as F(x).). So F(x)= -(1/2)e^(-(x^2)). The easiest way to do that is u substitution.

Now, lets integrate g(x). G(x)= x - sin(x).

Now that we know the two antiderivatives, we need to find out wher f(x) intersect with g(x); so where f(x) = g(x). Remember, this is a calculator portion, so you can simply plug this into a calculator (In fact, you can solve this question in a much easier manner using your calculator. I will show you how in my next post.). The results show the intersection, B, is at .94194.

Lets plug it all in! The integral of f(x) - g(x) from A=0 to B=.94194, can also be written as [F(x) - G(x)]0 .94194, which can be written as F(.94194)-F(0) - (G(.94194) - G(0)).

It all equals .591.

If you have any questions, please leave a comment.

Wednesday, January 3, 2007

Learn from the past AP Tests: 2000

AP (Advanced Placement) tests are supposed to test your knowledge of a specific subject to see whether or not you should deserve college credit for you knowledge. For calculus and physics, there is a multiple choice and a free-response section. Needless to say, if you can pass the test, it is safe to say that you have a very good grasp of the subject.

For those of you familiar with the AP tests, you know that they are very strict about their tests. They generally have you sign a paper that says you will never talk about the multiple choice questions and that you will not talk about the free-response questions till the questions have been published (which sometimes can take several years).

Fortunately, especially for those that will be taking an AP test in their future, some of the past free-response tests have been published for you to learn from. Now, without further adieu, here is the 2000 AP Free-Response Questions for Calculus AB.

There are two parts of the test, each spanning 45 minutes. The first section ONLY permits calculators. Of course, if you are looking at the test just for learning purposes, you do not need to follow such strict guidelines.

In future posts, I will take a look into some of the questions and try to help and perhaps give answers.

Monday, January 1, 2007

Happy New Years

Happy New Years!

Lets start the new year off with a bit of fun. Some of you may have heard this before.
  1. Pick a number 2-9
  2. Multiply your number by 9
  3. Add the two resulting digits together (For example, if you chose the number 4, multiplying it by 9 would result in 36. Then add the numbers 3 and 6 together.)
  4. Subtract 5 from your sum
  5. Now find the corresponding letter. If after you subtract 5, you are left with the number 1, your letter is "A." If you are left with 2, your letter is "B." And so on.
  6. Pick a country that starts with the letter you picked (ie. Argentina for A, Brazil for B)
  7. Now, take the last letter in the country you chose, and pick an animal that starts with that letter. (Brazil ends in an "L," so you might choose lizard)
  8. Now remember the country and animal.
Did you pick a Kangaroo from Denmark? :)

I hope you have a great new year.

Sunday, December 17, 2006

Get to know Isaac Newton

If you are at all interested in calculus or physics, you need to get to know Isaac Newton. The man was a pure genius. He created calculus just so he could do physics! Brilliant!

"Nature and nature's laws lay hid in night;
God said "Let Newton be" and all was light."
- Alexander Pope

Isaac Newton Wikipedia

Tuesday, December 12, 2006

Calculus 101: the website

I stumbled opon a great website today (while I was studying for a calculus final, actually). It is called Calc101.com. It is a great resource for beginning calculus students. It will differentiate and itegrate most functions for you. The site also has "graphs, matrices, determinants,
systems of linear equations, equations of lines" capabilities.

The site has been added to my links.