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

This implies

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

This implies

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

This implies

Since x'(t)=3, x(t)=3t. 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)=3t, 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

Kunquan Lan
Mon Jan 24 11:39:17 EST 2000