[Sample Problem]
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2015 IUT Math
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[Short Answer Type] [Sample Problem] Find a real number such that the limit lim → ∞
⋯
has a positive value.
Note: The actual value is equal to the integral
Find the area of the region of the -plane enclosed by the curve ln , the line , and the -axis. Note: The area is given by the integral ln
where ln . [Essay Type] [Sample Problem] For real numbers such that
, find the maximum and minimum values of the expression .
can be parametrized as cos sin for a real variable . Then, cos sin sin where is a constant satisfying sin
and cos
. Therefore, the given expression has the maximum value
and the minimum value . From the relation
, we may write ±
. Now the function ± has derivative ′ ∓ , which is zero when ± . That is when
, i.e., when ±
. At these critical points, is equal to ± ± . Therefore, the given expression has the maximum value and the minimum value . For a given constant , the equation represents a straight line with slope equal to . For this line to have an intersection with the circle , the range of the values of can be determined by finding the tangent lines to the circle with slope equal to . Using one of the various available formulas, the tangent lines have equations
± . Corresponding values of are ± . Therefore, the given expression has the maximum value and the minimum value
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