quiz3.nb

Calculus II	Quiz #3	February 27, 2002
Name____________________	R. Hammack	Score ______

(1) Use the Fundamental Theorem of Calculus to evaluate the following integrals. Simpify your answer as much as possible.
(a)

= 4+8-1+1 = 12

(b)

= 3(5) - 3 = 12

(c)

= sin(π/2)-sin(π/4) =

(d)

Because the integrand has an absolute value, we need to analyze it in some detail. Think of the graph of

It is a parabola that opens "up", and its x intercepts are 0 and 1. A graph based on this information is sketched on the right. As you can see,

is positive on the interval (-1,0) and negative on (0,1). Thus we break up our integral as follows:

1/3 +1/2 -1/3 + 1/2 = 1

(2) Find the area of the region under the graph of

between

and

(3) Suppose

Find the interval(s) on which F increases, and the interval(s) on which F decreases.
To answer this question we need to look at the sign of the derivative of F. By F.T.C. II,

.
The denominator of this expression is always positive, so the sign of F '(x) is controlled by the numerator.
As you can see, the nunerator is positive when x > 3, and negative when x < 3.
Therefore F(x) increases on the interval (3,∞), and decreases on (-∞, 3),