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

This implies

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.

Kunquan Lan
Wed Dec 1 13:53:22 EST 1999