AP Calculus

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Today we went over the homework, started looking at continuity, and took a quiz.

We went over the multiple-choice homework, specifically problems 3, 4, and 5 on the multiple-choice homework.

3. Remember that a negative exponent will become positive when it and its base are in the denominator. Therefore,   changes to . As x becomes larger, both the denominator and the numerator are increasing. However, the denominator increases at a greater rate than the numerator, meaning that as x gets larger, the limit goes toward 0.

4. Using the double angle formula, the numerator,  , is realized to be equal to  . This is put over  , which cancels out with the   in the numerator. Therefore,   is equal to  . This oscillates from -2 to 2; therefore there is no limit.

5. In order to solve this problem, do not multiple by the conjugate (because x is approaching infinity). Instead, multiply by   in order to get the   in the denominator (in either the numerator or denominator of the larger equation). It now looks like this:  . Because there are   in the denominators of the fractions, as x gets larger, the fraction will move toward 0 (such as the example  ). Therefore, it becomes:  .

After reviewing the homework, the new idea of continuity was introduced. Continuity (at x=c) means that one can get from point A to point B on a graph without leaving the graph. The vocabulary word cusp was learned (meaning point or apex).
In order to have continuity, the graph/point must have all three qualities listed below:
- f(c) must exist (no hole, asymptote, etc.)
-   must exist
-  = f(c) (most important!)
For example, there may be a point at (2, 5), but the limit of f(2) may be 3.

Discontinuity and removable discontinuity were also explored.

Exploration 2-4 was done.

There was a quiz at the end of class.

The homework on pg. 56 are all Q’s and 1-65 (odd)

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