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

NAME:

Student Number:tex2html_wrap_inline146 No.tex2html_wrap_inline148Marks tex2html_wrap_inline150


tex2html_wrap_inline152 (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.

1.gif (2102 bytes)

(2). Hypotheses: x'(t)=3, |AB|=5 and tex2html_wrap_inline168.

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

Solution: By the above figure, we have
 equation29

Taking the derivative of Eq. tex2html_wrap_inline176 relative to t, we have
displaymath180

This implies
 equation36

By Eq. tex2html_wrap_inline182, we have tex2html_wrap_inline184. This, together with Eq. tex2html_wrap_inline186, implies
displaymath188

When y(t)=3, we have tex2html_wrap_inline192.

tex2html_wrap_inline194 (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) tex2html_wrap_inline198.

2.gif (22.gif

Solution:

tex2html_wrap_inline200 and tex2html_wrap_inline202. It follows that
displaymath204

(b) tex2html_wrap_inline208, where tex2html_wrap_inline210

Solution:

3.gif (43.gif

tex2html_wrap_inline212 and tex2html_wrap_inline214 Hence, we have
displaymath216

tex2html_wrap_inline218. (5 Marks) Find the area of the region enclosed between the graphs of the functions
displaymath220

Solution: Let tex2html_wrap_inline222. Then tex2html_wrap_inline224. Hence, x=-2, x=0 and x=2.
eqnarray100


eqnarray114

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


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