AP Calculus

More Work With Limits:

We reviewed and answered Concepts Problems on pg. 35:

    ”The derivative of f(x) is equal to, which is the average rate of change and, therefore, the slope of the secant line.”

      - To solve for the instantaneous rate of change (derivative) using the equation above you must:

          1.  find the value of f(c) and then subtract it from the equation of f(x)

          2. factor the equation in order to divide by x-c (note: if the equation is not easily factorable it may be necessary to use synthetic division to divide the equation f(x) by x-c)

          3. divide by x-c to find the equation for the limit

          4. plug the value of c into the equation for the limit in order to find the derivative

          * see page 35 Concepts Problems: C1 for an example of how to solve this with values

We reviewed Exploration 2-2:

     Things to remember with finding delta:

        - always use the smallest value of delta when saying what d equals in the definition for limit

                ex: problems 2 & 3 the potential values of delta determined are 2.6 units and .9 units. In declaring the value of delta always use the smaller value: in this case, .9 units.

        - to find the exact value of delta when the epsilon and equation are known you must set f(x) equal to the limit + and - the epsilon. For example, if the limit was 6 and the epsilon was 1, you would solve for x in two equations, one where f(x) = 7 and one where f(x) = 5.

              - then subtract the value of c from the determined values of x to find the delta (and again, choose the smallest value of delta for the definition)

Additional Class Notes:

     Limit Terminology Example: (where Epsilon = .5, Limit = 5, and C = 3)

            ”if and only if for any number epsilon = .5 there is a number delta > 0 such that for any x within delta units of 3, then f(x) is within .5 units of 5″

     Properties of Limits:

      - if:then:

              -

              -

              -

              -(when x ≠ c)

There will be a quiz of Friday, September 26 – you will need to know:

Homework for Wednesday, September 24:

Pg. 45: problems 7, 9, 11

“Investigating Limits on the TI-84 or TI-89 Calculator:” problems 1-5

** Also: for extra credit turn in correct answer to question 8 on exploration 2-2 by Friday, September 26

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