So the tea act and stamp act and stuff was all part of the Townshend acts?
I believe so
Yeah they were more than one law. But the whole idea of learning about the townshend acts is usually to show how the British taxed so many ridiculous things without the consent of the people, and that made everyone really mad. After much discussion, parliament took away most of the taxes, but kept the tax on Tea, which lead to the Boston Tea Party.
Has anyone here taken calculus, or is taking it? Because if you have I need help lol
I’m in Stats right now but I’ll take stats 2 or pre-calc next year!
I had several semesters of calculus but it’s been a long time.
AP CALCULUS BC IM HERE TO HELP!
Are you planning on taking the AP test at the end of the year?
YAY! I’m in AP calc AB
Have you done finding limits algebraically? My teacher explained it to us quickly, but I didn’t really get it…do you have a simple way of explaining it?
Yup yup! I took the AB test last year too.
Finding limits algebraically? Yeah. Give me a moment to grab my notes and then I’ll be right back with a picture. Cause there’s like… three different ways to solve algebraically, but they all come back down to one thing.
Alright, so the first step is called “Direct substitution”. That’s when you plug the number the limit is going to straight into the equation, which works for equations like x^2, but become problematic for equations that end up as 0/0.
For equations like the above, you have three choices (four really but I couldn’t care less for the fourth. It’s useless and can always be found another way).
Your options are: Factor and cancel, Rationalize and cancel, and common denominator.
Factor and Cancel will work best for this equation. (x^2 - 9) factors into (x + 3) and (x - 3)
Now you can substitute in 3 for x (direct substitution again) and get 6 as your limit.
The basic idea of factor and cancel is literally that: factor either the numerator or denominator, and cancel out any unnecessary expressions.
(End of Part 1. I don’t want to overwhelm you so any questions?)
Ohhh ok, thanks! I kept getting 0/0 for almost all of my answers and I knew that couldn’t be right. No questions yet, I think I understand factor and cancel.
Alright, the next one is rationalize and cancel.
Let’s take this equation:
As you can see, the answer when you do direct substitution is 0/0, and that’s not what we want.
Now. This is where most of the mistakes are made. What you want to do, is get rid of the square root, so to do that, you must multiply the numerator’s equation by a fractional 1. Numerator and denominator must be the OPPOSITE equation as the numerator. It’s hard to explain, so here’s a picture. (as shown):
This leaves you with a new numerator. Don’t multiply out the denominator.
As you can see, you can now cancel out the (x-4) from both numerator and denominator. After another direct substitution, you get: 1/4
No questions, what you said makes sense. The pictures help Thanks!
YAY! Last one before I’m just going to link you to a Squeeze Theorem video because I hate explaining it.
Last one is Common Denominator. Use this one when you have a fraction for a numerator.
Here’s our equation:
What we need to do is get rid of the double fraction.
So let’s multiply the numerator by the common denominator! Again, use a fractional one, and do a separate equation for each expression on the top. Remember to distribute the subtraction sign if you were originally subtracting the equations.
Because of fraction rules, you can multiply the numerator by (1/denominator), or in this case (1/x). From there, you can cancel the 'x’s and get the new equation. Use direct substitution, and you get: -1/49
thanks for giving me advanced education. this might help in a few years.