MATH 1300.03F D **Exam 3** Wednesday, December, 1 1999

NAME:

Student Number: No.Marks

(5 Marks) A 5 meters long ladder is leaning against a building. The bottom of the ladder is dragged along the ground, away from the building, at 3 meters per minute. How fast is the top of the ladder moving down the side of the building when it is 3 meters above the ground?

Solution: (1). We draw the following figure.

(2). Hypotheses: *x*'(*t*)=3, |*AB*|=5 and .

3. Question: Find *y*'(*t*) when *y*(*t*)=3.

Solution: By the above figure, we have

Taking the derivative of Eq. relative to *t*, we have

By Eq. , we have . This, together with Eq. , implies

When *y*(*t*)=3, we have .

(5 Marks) Sketch the rough graph of the area given by each of the following definite integrals. Use this graph to determine the value of the definite integral.

(*a*) .

Solution:

and . It follows that

(*b*) , where

Solution:

and Hence, we have

.
(5 Marks) Find the area of the region enclosed between the graphs of the functions

Solution: Let . Then . Hence, *x*=-2, *x*=0 and *x*=2.

Hence, the area of the region =|*A*|+|*B*|=4+4=8.

Wed Dec 1 13:53:22 EST 1999