There are three parts to this assignment.

Part A is to be completed online before 7:00 a.m. on Friday,

October 29.

Part B and Part C , which require full solutions, are to be handed in at the beginning of class on the same date.

Part A [10 marks]

This part of the assignment can be found online, labelled A5 , at

— sign in using the MATH110 001 2010W button.

Part B [5 marks]

This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions.

From the Calculus: Early Transcendentals text, complete questions 48 and 72 from section 3.1.

Part C [15 marks]

This part of the assignment consists of more challenging questions. You are expected to provide full solutions with complete arguments and justifications.

1. Imagine a circle anywhere in the universe. Prove that there are two points directly opposite each other on the circle with the same temperature. (Hint: assign an angle θ to each of the points on the circle, and let T ( θ ) be the temperature at that point. Then apply the Intermediate Value Theorem to the function f ( θ ) = T ( θ ) − T ( θ + π ).)

2. Find two tangent lines to the curve y = x

2 which pass through the point (2 , 3).

3. Write down the expression of a function which is continuous everywhere, and differentiable everywhere but at x = 1 , 2 and 3.