Biweekly MAT157 Writing Challenge
The following excerpt is trying to convey mathematical ideas. We challenge you to explain those ideas clearly and concisely. Here is the excerpt for the week of March 21, 2011:
There are two numbers that are both positive. One is bigger than the other. The bigger one is even. The smaller one is odd. The absolute value of their difference is smaller than either number and is, consequently, odd. The three numbers, arranged in ascending order, when plotted on the x-y-z axes, will be located in the region where all coordinates are positive. These numbers can be given the names a and b (for the original numbers) and c for their difference.
Send your work to writing@math.toronto.edu by noon on Friday to receive bonus marks for this weeks challenge. The best response will be posted in the archive.
Last excerpt:
When you graph f(x), it goes to both positive infinity and negative infinity from the left and right respectively when x gets really close to 2. But everywhere else, f(x) is defined and finite and smooth and has no breaks. If you shift f(x) up and down, it continues to have these properties. In fact, by shifting it up and down you can create a family of functions that all have these properties.
Solution:
Let f(x) be a differentiable function on\ {2} that goes to positive infinity as x approaches 2 from the left and negative infinity as x approaches 2 from the right. Then ft(x) = f(x) + t defines a family of smooth functions on the same domain.