If , evaluate .

Difficulty Rating:

Evaluating this limit is essentially equivalent to finding the derivative (which we know to be 2x, according to the Power Rule). To find the limit, plug the two quantities indicated in the numerator into the function f(x). Then, find the limit by factoring:

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Calculus II (BC): Parametric Equations

Let P be the curve defined by the parametric equations .
        (a)    Graph the portion of the curve from  (without a calculator).
        (b)    Convert to a rectangular equation.
        (c)    Find the slope of the tangent line when . Difficulty Rating:

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Solution:

    (a)    To graph, you should plug in a number of values of theta for the indicated interval. Your graph should look something like this:

    (b)    Remember the trigonometric formula , and this problem is quite easy. All you have to do is solve the parametric equations given for x and y and plug them into the formula.     (c)    The derivative, , of a parametric curve is given by . Findind the derivatives of x and y with respect to theta is very easy; once you calculate them, plug them into the formula for dy/dx: Now, plug in  to find the requested derivative:

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