University of Toronto
Edward S. Rogers Sr. Dept.
of Electrical & Computer Engineering

Tuesday, March 5, 2002

MAT197S - Midterm Test

Duration: 2 hours     6:00 - 8:00 pm

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Last Name: ____________________________________

First Name: ____________________________________

Student No: ____________________________________

Tutorial No: ____________________________________

No calculators or any aids are allowed.

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1. Evaluate

(a)

(b)

(c)

2. Solve the initial value problems:

(a)

(b)

3. Find the length of the curve defined parametrically by

     

for

4. A lake with a volume of 8 billion ft³ contains pollutant in a concentration of 0.25 u/ft³ water daily with a pollutant concentration of 0.05 u/ft³ removes 500 million ft³of the lake water daily. Assume that the water in the lake is perfectly mixed at all times. How long will it take to reduce the pollutant concentration in the lake to 0.10?
*This question was very strangely worded on the midterm and had to be clarified by the professor. I can't remember how it was clarified though. Try your best.

5. (a) Show that the curve with parametric equations intersects itself at the point (3,1), and find equations for the two tangent lines to the curve at the point of intersection.

(b) Find at the point (5,9).

6. (a) Find the area above the line and inside the circle r = 2.

(b) Let and . Find .