MATH 1014.03MW **Exam 1** Monday, January, 24 2000

NAME:

Student Number: No.Marks

**Instructions**: This exam contains 4 questions and has a total of 100 marks. Show
all of your work.

(30 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 (*i*). We draw the following figure.

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

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

(**Method 1**) 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 *x*'(*t*)=3 and *y*(*t*)=3, we have .

(**Method 2**) 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 *x*'(*t*)=3 and *y*(*t*)=3, we have .

(**Method 3**) By the above figure, we have

Since *x*'(*t*)=3, *x*(*t*)=3*t*. This, together with Eq. ,
implies

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

When *y*(*t*)=3, by Eq. , we have . Since *x*(*t*)=3*t*,
*t*=*x*(*t*)/3=4/3. Hence, when *y*(*t*)=3, that is, *t*=4/3,
we have

(25 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. We draw the following graph.

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

(25 Marks)Find the volume of the solid generated by revolving the region
bounded by the parabolas and about the *x*-axis.

Solution: Let . Then *x*=0 and *x*=1. We draw the following graph.

(**Method 1**)

Region: and for .

Axis of revolution: *x*-axis.

The volume .

(**Method 2**) Region: and , .

Axis of revolution: *x*-axis.

The volume .

(20 Marks) Write a sum of two integrals which equals the area of the region inside and .

(You need not evaluate the integrals).

Solution: (*i*) We draw the following graph.

(*ii*) Find the points of intersection.

Let . Then This implies

Hence, or . So, we obtain or .

(*iii*) Using the formulas of the areas of polar equations, we have

Mon Jan 24 11:39:17 EST 2000