***MTH 111 EXTRA CREDIT PROBLEMS***
All extra credit problems are due anytime before the end of the term. They can be typed or handwritten.
Please be thorough when writing up your solutions and show all of your work/give complete answers,
depending on the problem.
EC #1: Worth 5 points toward your overall homework grade
Read about "completing the square" in section 3.2. Write up a complete solution to problem #114 on
page 205. Follow the step-by-step instructions given to you in order to derive the quadratic formula.
EC #2: Worth 5 points toward your overall test grade
It turns out that a standard 12 inch ruler has more inch markings than it needs. You can take some
markings off and still measure everything up to 12 inches. Here’re some examples to show you what I
mean.
To measure all of the lengths between one and three inches, you only need one mark on a three inch
long ruler. You can measure one inch between the 0 and 1, two inches between the 1 and 3, and three
inches between the 0 and 3.
0 1 3
To measure all of the lengths between one and six inches, you only need two marks on a six inch long
ruler. You can measure one inch between the 0 and the 1, two inches between the 4 and the 6, three
inches between the 1 and the 4, four inches between the 0 and the 4, five inches between the 1 and the
6, and six inches between the 0 and the 6.
0 1 4 6
Your job for this extra credit problem is to figure out the minimum number of marks needed on a 12
inch ruler in order to measure all of the lengths between one and twelve inches. Your answer should
include a drawing of the ruler with the necessary labeled marks on it as well as an explanation for why
your ruler works and why you think your number of marks is the minimum. There can be multiple
correct ruler configurations.
EC #3: Worth 5 points toward your grade on the final exam
Consider the function f(x) = ix, where I is the imaginary unit we learned about in class.
1. Evaluate the following expressions:
* f(1)
*f(f(1))
*f(f(f(1)))
*f(f(f(f(1))))
2. What happens when you continue this process and compose the function with itself repeatedly?
What pattern emerges from this process?
3. Pick another complex number (a number of the form a + bi, where a and b are real numbers) and
make that number your starting input instead of 1. Does the same pattern hold?