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

Monday, November 5, 2001

MAT196F - 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. Does the function

(a) have a derivative at x = 0? Explain. (3 marks)
(b) have a derivative for all x? Explain. (3 marks)
(c) What is ? (2 marks)

2. Show that if the cubic polynomial has a local minimum and a local maximum, then the point of inflection is the midpoint of the line segment connecting the local extrema on the graph. (7 marks)

3. Sketch the graph of

(by determining the domain, the asymptotes, x and y-intercepts, concavity, increasing and decreasing, ...). (8 marks)

4. An airplane, flying at 450 km/hr at a constant altitude of 5 km, is approaching a camera mounted in the ground. Let be the angle at which the camera is pointed when . How fast does the camera have to rotate in order to keep the plane in view? (7 marks)

5. Prove that the following limit exists using the definition of a limit. (8 marks)

6. (a) Which point on the graph of is nearest to the point (0,3)? (5 marks)

(b) Evaluate:

(2 marks) (i)
(2 marks) (ii)

(c) . (3 marks)