Part 1: Solve the Basic Mixture Problem

(1) Solve the following mixture problem, using any method(s) you wish:

Consider two containers, 1 and 2, containing 100 cc of solution a and solution b, respectively. Ten cc of solution a is taken from container 1 and placed in container 2. The mixture in container 2 is then mixed up and 10 cc of this blend is placed in container 1. Determine if there is then more of solution a in container 2 or more of solution b in container 1.

(2) Share with the class your thinking about this problem, and demonstrate and justify your method(s) of solution. Comment on the strategies you used and their effectiveness.

(3) Compare and contrast your thinking and methods with those of other class participants.

(4) If not determined as part of your solution above, find the amount of solution a in container 2 and the amount of solution b in container 1. Record these amounts.

(5) Suppose that after the solutions are mixed, as described above, the process is carried out again. That is, 10cc are taken from the resulting mixture in container 1 and are placed in container 2 and blended. Then, 10cc of this new mixture in container 2 is put into container 1. Again, determine if there is now more of solution a in container 2 or more of solution b in container 1. Record the amount of solution a there is in each container after each full iteration of the process (note: a full iteration involves two "scooping" actions).

(6) Predict the amount of solution a there will be in containers 1 and 2 if this process is continued indefinitely.

(7) Generate a pair of recursive equations for the amount of solution a in each container at a given state, in terms of the amounts in the containers at the previous state. Discuss how you derived these equations.

(8) Enter the recursive equations in your graphing calculator and generate tables of values for the amount of solution a in each container after many interactions of the mixing process.

(9) Plot these values. Discuss your observations.

Part 2: Exploring Continued Fractions

(10) Using algebraic methods, determine the value of the continued fraction:

(11) Derive a recursive equation to generate the value of this continued fraction.

(12) Enter this recursive equation in your calculator and determine the value of the continued fraction recursively. Plot the sequence of values generated and observe the graph.

(13) Comment on the different approaches used to determine the value.

Part 3: Fibonacci Sequences

(14) Generate the 1st 10 terms of a Fibonacci Sequence, starting with initial values 1 and 1.

(15) Derive a recursive expression to generate this series, enter the equation in your calculator, and generate the series recursively.

(16) Generate the 1st few Fibonacci Ratios (ratios of successive Fibonacci terms, e.g. 1/1, 2/1).

(17) Derive a recursive equation to generate a sequence of Fibonacci Ratios, enter this equation in your calculator, generate a sequence of ratios, and plot this sequence. Determine the limit of this sequence.

(18) Generate several more sequences of Fibonacci Ratios, starting with different initial values, and plot these sequences. What do you observe, numerically and graphically, about the limits of these sequences?

(19) Conjecture a value for this limit, and prove your conjecture.

Extension

Generate and plot several sequences of Fibonacci Ratios with a spreadsheet. Comment on the differences between using a graphing calculator and a spreadsheet.

Part 4: Solving Equations Recursively

(a) Discuss solutions to the equation: X = cos(X).
(b) Solve this equation graphically.
(c) Solve this equation recursively.
(d) Comment on the different approaches.