Lesson from September 17

Question of the Day:

Find the equation of the horizontal asymptote for the given function:

Factor the x^4 out of the numerator.

Multiply the entire equation by 1 by mutliplying the numerator and denominator each by 1 divided by the highest degree of x in the denominator.

Distribute the 1/(x^2). Also, recognize that the (2/x^4) in the numerator has been replaced by 0 because it approaches 0 as x gets closer to infinity.

Recognize that (x^2/x^2) in both the numerator and the denominator are equal to 1. Also, recognize that (4/x^2) has been replaced by 0 because it approaches 0 as x gets closer to infinity.

Simplify.

Therefore, the horizontal asymptote is y= -1.

Now onto the lesson. We learned the Intermediate Value Theorem and it's Corollary, and to do so, we performed the following instructions.

-On a graph, draw a function between two endpoints

-On that line, choose two points 'a' and 'b', and label them.
-Find f(a) and f(b).

-Define any point W between f(a) and f(b).

Intermediate Value Theorem:

Hypothesis:
-Suppose f is continuous on the closed inverval [a,b].

-W is any number between f(a) and f(b).

Conclusion:

-There exists as solution within [a,b] such that f(c)=W.

-There is at least one x value for W, your chosen y value.

-If f(x) is continuous, it can't skip any values.

Corollary to the Intermediate Value Theorem:

Hypothesis:

-Suppose f is continuous on a closed interval [a,b].

-f(a) and f(b) have opposite signs.

-[This is true of the above graph].

Conclusion:

-There is at least one number within [a,b] such that f(c)=0.

-The function must cross the x axis at least once, so there is at least one zero.

So, that's about it. Hurrah for finally figuring out how to work this thing. If anybody has any questions, come talk to me and I'll try to explain it better. Good luck studying for the quiz.