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MA 303 Homework 1 1) The TB-12 bacteria has a daily growth rate of 10%

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MA 303 Homework 1
1) The TB-12 bacteria has a daily growth rate of 10%. 
a) Write a difference equation to model the number of bacteria in a Petri dish with xo TB-12 bacteria at the beginning of the first day. 
b) Use Excel (or Google Sheets) to numerically solve the difference equa-tion with xo = 2 for xj, x2, ..., x20. 
c) Make a scatterplot of the solution, with n on the x-axis and x„ on the y-axis. Send me the file with the scatterplot. 
d) As n gets large, what happens to xn? Explain why this model for the bacteria might not make sense for large n. 
2. Each minute, an eggplant placed in a 400 degree oven heats up by one-fifteenth of the difference between 400 degrees and it current temperature. a) What is An? 
b) Using part (a), write down a difference equation that models the tem-perature after n minutes in the oven. 
c) Use Excel (or Google Sheets) to ntunerically solve the difference equa-tion. Use this to find the temperature of the eggplant after 40 minutes in the oven, if the original temperature of the eggplant was 65 degrees. 3. At the end of each year, a bank account adds 4% interest to the account balance. Immediately after interest is added to the account, you withdraw $100. 
a) Write a difference equation to represent the account value after n years of this process. 
b) If the account begins with $1200, how many years will it take for you to wipe out the account? You should use Excel or Sheets. 
4. Make a conjecture xn = g(n) for the analytical solution to the difference equation in question 1. Use the principle of mathematical induction to prove that your solution g(n) is correct.