AP Calculus AB and BC: Review #1 for the AP Test

    If 

    and g(x) = 2,

(a)    Find the value of k which makes f continuous.
(b)    If f is continuous, find the area in the first quadrant bounded by f,g, and x = 4.

Difficulty Rating: 

(a)    If f is continuous, then a limit must exist at x = / 2. In essence, this means that the cosine piece of the graph and the parabola piece must meet at x = / 2. To meet, they must have the same function value when / 2 is plugged in to each; this ensures both pieces reach the same height there and there is no break in the graph. Below is the resulting graph of f. The blue piece is the portion defined by cosine and the red is defined by the newly-defined parabola. Note that they both contain the point (/ 2, 0) as we intended so the graph is continuous. (b)    In order to find the bounded area, you'll need to use two separate integrals, as the function is defined by two other functions. This is easy with a calculator and simple (albeit messy) by hand. The answer according to the calculator (I wussed out, no doubt about it) is 15.048.