AP Calculus AP Demonstration

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AP CalculusAP Demonstration

• 2005 AP test Question 4

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• Let be a function that is continuous on the
  interval 0,4). The function is twice
  differentiable except at x 2. The function
  and its derivatives have the properties indicated
  in the table above, where DNE indicates that the
  derivatives of do not exist at 2.

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For 0ltxlt4, find all values of x as which has a
relative extremum. Determine whether has a
relative max or min at each of these values.
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O.K., is zero or DNE at 1 and 2, so those are
the relative extrema. The slope goes from
positive to negative at 2, so there is a max at x
2.
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On the axes provided, sketch a graph of a
function that has all the characteristics of .
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WOW!
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Let g be the function defined by g (x) the
integral from 1 to x of (t) on the open interval
(0,4). For 0 lt x lt 4, find all values of x at
which g has a relative extremum. Determine
whether g has a max or min at each of these
values.
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g (x) has a relative extrema at x 1 and at x
3. There is a min at x 1, because the area
under f(x) is decreasing, then increasing, and a
max at x 3 because the area is increasing, then
decreasing.
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For the function g (x), find all values of x, for
0 lt x lt4, at which the graph of g (x) has a point
of inflection!
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g (x) has a point of inflection at x 2, because
at that point the area under the curve of f(x)
goes from increasing at a faster rate, to
increasing at a slower rate!
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Thats it! Man, that was easy! Im Outta here!