A video about taking AP courses in high school

Here's a video about students who take AP classes in high school. It is from the New York Times. You can find it here. How do you feel about this? Leave a comment if you wish.

Mr. Corn

Precalculus 1/27/10

Sinusoidal functions - day 2

Today we learned how to move the graph of sine up and down. This is called vertical shift. What's important about this is that you drawn in what's called a sinusoidal axis to help you graph the function. Here's an example...







As you might expect, the graph is translated up 2 units. We say this has a vertical shift upward of 2. The range of this graph is [0,2]








Here's another example where the graph is translated in a different way.










This graph is moved pi/2 to the right.

This is an example of phase shift of positive pi/2. If the graph is translated to the left, the phase shift would be negative.









Some very helpful hints...

If you have the general sinusoidal equation y = Asin(Bx+C)+D...

1. A is the amplitude
2. The period of the function can be calculated by



3. The Phase shift can be calculated by




4. The vertical shift is D

Hope this helps. You can find more notes on Mr. Corn's webpage here.

Pre-Calculus 1/26

Today we learned how to find the sin(x), and what that looks like on a graph.

We found the (sin) of five different points using our Unit Circles:

We then used those points to make a graph:

We then found the domain, range, and period for the graph:

(the Period is to what (x) plot the data goes)

Domain: All Real Numbers

Range: [-1,1]

Period: 2(pi)

We also learned how to graph 2sin(x):

Domain: All Real Numbers

Range: [-2,2]

Period: 2(pi)

As you can see, the range became bigger. This increases the graph's AMPLITUDE.












Edited by Mr. Corn....

Let's also look at a graph where the variable is multiplied by a constant....

y = sin(2x)



The red graph represents the graph of sin(2x). It's period is double of the base graph of sin(x) in blue.

The period of the red graph is pi

The period of the blue graph is 2pi






Helpful hints:

1. To find amplitude, take the absolute value of the coefficient in front of sine. For example, the amplitude of y = -3sin(x) is 3.

2. To find the period of y = sin(bx), calculate 2pi divided by b.

3. When drawing these graphs by hand, ALWAYS list your x values on the x axis. Examples of this can be found on Mr. Corn website here.

Pre-Calculus 1/25

What we learned today was how to find trig values that go around the unit circle more than once.




I realized that 9 pi over 4 wasn't on the unit circle. So, you have to subtract 2 pi from 9 pi over 4 because that how much it is to round the circle one time.




You have to get a least common denominator. Then subtract and find the value on the unit circle.

Here is another example:








Here are the equations we learned in class to help find the negative radians on the unit circle.







Here are some examples:













Ben's Beliefs:
1. know that 2 pi is once around the unit circle
2. you may have to go more than once around the unit circle to find your answer
3. knowing the unit circle will help in the long run....
4. "And in the end, the love you take is equal to the love you make" - The Beatles

you can find the rest of the notes on my dad's web page here

Precalculus 1-21-10

Unit Circle Day 2

Today, we all learned how to find the value of different problems from the unit circle.
Remember: Change your calculator settings to Radians instead of Degree because the problems won't work otherwise.
1)

To solve this problem, all you need to do is type into your calculator. The answer should be .7431. All answers should usually be rounded to four digits.


2)

I know you all are probably frowning right now because in this problem, there isn't sin, cos or tan which are the no brainers. But, this problem is simple as well! On the unit circle,
cot=. To solve this problem, we are going to use the same formula from the unit circle to get our answer. So, you take the formula and just plug in the numbers 2 and 9 from the original problem. Your final formula will be . Just plug that formula into the calculator and the answer will be 1.1917. :)


3)

There's no pi in this problem, so we can't do it right? Wrong! We all know that in the unit circle, . To solve this problem, all you need to do is what we did in the previous problem. Take the 5 from sec5 and plug in into the formula . Your final formula should be
. Punch the formula into your calculator and your answer should be 3.5253.


I hope these questions help you guys if you are having trouble. If you need any more help, then go onto Mr.Corn's Website Here .

Precalculus 1/20/10

Hey guys, it's Peyton with the first student post on the blog.

Today we learned how to use the unit circle. We also learned some equations that helped us utilize the unit circle.

Here are some of the equations we didn't learn in class:

Let's do some examples.

First, we'll see that pi over four is a first quartile 45-45-90 triangle. Looking at the equation for sin above, we know that we only need the y value. So, our answer is:

Just like the last problem, we'll first find the type of triangle we're dealing with. pi over three is a 60-30-90 right triangle. For cosine we need the x value, so our answer will be

Let's do one more example and mix it up a bit.

Before we flip out because its cotangent, let's remember that it's simply the reciprocal of tangent. First, we find that this is a 30-60-90 triangle. What is different is that it is in the second quartile, which means its x value is negative. Just put the x value on top of the y
and we get our answer:

But we're not through yet. Remember that we cannot have a fraction divided by a fraction. Negative square root of three and one are both being divided by two, we can cancel them out. This gives us our true answer

Peyton's Points: Here's some tips when doing problems with the unit circle.

1. Find out what kind of triangle you're being asked to solve.
2. Remember to think about which quartile it's in.
3. Remember that co secant, secant, and cotangent are all reciprocals of our basic sin, cosine, and tangent.
4. Rationalize your denominator if necessary.
5. Remember, you can't have fractions divided by fractions. See if you can cancel out the denominators of the fractions.

Woohoo! First post is in the books. If you need any more help, check out Mr. Corn's Website. Until next time, keep your pants off the ground, or you WILL be looking like a fool.

Peyton

Precalculus 1/19/10

Today we learned how to solve some science applications using exponential formulas such as the Law of Uninhibited Growth and Newtons Law of Cooling. Here's an example.

Example #1 Law of Uninhibited Growth

A culture of 2000 bacteria is growing in a petri dish. There are 2200 bacteria after 1 hour. Using the Law of Uninhibted growth, how many are there after 8 hours? 1day?

The Law of Uninhibited Growth

stands for the initial amount. Other values get substituted. Now solve for k


get e by itself


take natural log of both sides


solve for k. Make sure you store the variable



substitute back into original formula. Use inititial amount and t = 8

A=4287

When t = 24 (because there is 24 hours in a day)



When will there be 10,000 bacteria?


substitute 10,ooo in for A. Use the same initial value and the same k

divide both sides by 2000




Example #2 Using Newton's Law of Cooling

Here's an example that involves Newton's Law of Cooling. The formula that we'll use for this is the following:



u(t) represent the Temperature of the object after a certain time
T represent the temperature of the surrounding area (usually room temperature)
represents the initial temperature of the object

t represents the time

Just like the previous problem, we'll have to find k first, rewrite the equation, then answer the question.

Now here's an example...

An object is heated to 100 degrees. The object is 80 degrees after 5 minutes. The temperature of the room is 30? When will the temperature of the object be 50 degrees?

OK, let's find k first by substituting everything we know...

substituting...

parentheses first..

get e by itself, take the natural log of both sides


now solve for k, but store k in your calculator!


now rewrite the formula, now we are going to solve for t, with u(t)=50

solve for k


Helpful hints:

Always solve for k first, then rewrite the formula, then answer the question
You need to memorize then formula for the law of uninhibited growth, but not Newton's Law of Cooling.

You can find other examples at Mr. Corn's web page here.