Class notes for 10/9/07
- Today we started off with a review of closed intervalsIn one of the homework problems we were asked a question relating to an inclusive inequality notation. The problem had the notation [-1, 3] in it, the inequality is
- When the notation is in the form of (-1, 3) then the interval reads as
When mixing ( and [ The inequality changes for that side of the x. The parenthesis are less than and the brackets are less than or equal to.
The second thing we did in class is learn about the Intermediate Value theorem
The definition is on page 67 of the text book: "If a function is continuous for all x in the closed interval [a, b] and y is a number between f(a) and f(b), then there is a number x=c in (a, b) for which f(c) = y.
Continuity is necessary for this theorem to be true.
This basically says that between the two f(x) values for any value of x, a y value can be chosen and when the x is solved for, the value will fall between the two x values. The value chosen is on the y axis, so you will need to find the y values for the x’s given.
For this equation f(x)=2x+3 I chose the x values 2 and 5. For my y value I chose 9, which is between the f(2) and f(5). The x value was 3 , so it fell between the x values 2 and 5, proving the theorum.
There is a calculus phobe video on the theorem, linked off Mrs. Evrard’s website.
For the rest of class we did practice on problems 3&4 on page 68
The homework was: pg 68 all quicks, pgs 74-76 T2-5, 7, 9, 10, & 11
October 11th, 2007 at 5:53 pm
It’s not that continuity is necessary for the conclusion to be true, (remember the examples where the conclusion was true but the function was not continuous?) it’s the fact that if you have continuity them the conclusion MUST be true.