1. Linford Group Meeting Department of Chemistry and Biochemistry Brigham Young University
Thursday, Feb. 23, 2017
Problems on the oxidation of tertiary amines, the Cope elimination, the triangle function, and the Bisection Method for finding roots of equations
by Matthew Linford

2. Chemistry
What happens when I treat the following tertiary amine with mCPBA? Hint: assuming R1 – R4 are alkyl/aryl groups, where is the molecule most likely to oxidize?
Draw a mechanism for the following reaction (the Cope Elimination). Note the stereochemistry. What is the structure of the byproduct that is not drawn?
mCPBA

3. Math The triangle function is defined as follows:
Λ(x) = 0 for |x| > 1 and
1 - |x| for |x| < 1
Graph Λ(x). What is the area of Λ(x)?
What is lim τ  0 Λ(x/τ). What is the area of this function? This is an example of what kind of function?
What is lim τ  0 (1/τ) Λ(x/τ). What is the area of this function? This is an example of what kind of function?

4. Math
An important problem in numerical analysis is that of finding roots of equations. (A ‘root’ is a point where a function crosses the x-axis.) The simplest method for determining the roots of a function, f(x), is the bisection method. We’ll start with this method/algorithm.
The bisection method is based on the Intermediate Value Theorem (IVT). This theorem says that if f ε C[a,b], i.e., if f, the function, is continuous over the closed interval that lies between a and b, and K is any number between f(a) and f(b), then there exists a c in [a,b] for which f(c) = K. Draw a picture that explains this theorem, and convince yourself that it is reasonable.

5. Math The bisection method says:
Suppose a continuous function is defined on [a,b] and that f(a) and f(b) have opposite signs.
By the IVT, there exists a p, a < p < b for which f(p) = 0. The bisection method consists of repeatedly halving the subinterval [a,b] and at each step locating the ‘half’ containing p.
Use the bisection method to find a root of f(x) = 3x4 – 2x2 – 4 over [1,2]. Find this root to an accuracy of 10-2.
In the above definition of the Bisection Method, could we replace the statement: “f(a) and f(b) have opposite signs” with “f(a)*f(b) < 0”? Where might this new definition be useful?

6. Math
Derive an equation that gives the accuracy (error) of the Bisection Method, or at least an upper limit for its accuracy, in terms of n, the number of steps (bisections) made.
Calculate the number of steps (bisections) you would have to take to guarantee an error less than 10-5 for the problem we just solved:
Use the bisection method to find a root of f(x) = 3x4 – 2x2 – 4 over [1,2]. Find the root to an accuracy of 10-2.
What can you see as an important advantage and an important disadvantage of the Bisection Method?