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Additional resources for Analysis, Calculus. Difference-eq To Differential eq
13. The first example of this section is a particular case of the more general problem of computing probabilities associated with the waiting time for some event to occur. As another example, suppose that an electronic switch works with probability p and fails with probability q = 1 − p. Then, using reasoning analogous to that used in the coin tossing example, the probability that the first failure will occur on the nth use of the switch is pn−1 q, n = 1, 2, 3, . .. (a) Can you justify this probability?
Let wn be the number of weasels n years from now and suppose that there are currently 350 weasels. (a) Write a difference equation which describes how the population changes from year to year. (b) Solve the difference equation of part (a). If the population growth continues at the rate of 3%, how many weasels will there be 15 years from now? (c) Plot wn versus n for n = 0, 1, 2, . . , 100. (d) How many years will it take for the population to double? (e) Find lim wn . What does this say about the long-term size of the population?
A glass of lemonade, initially at a temperature of 42◦ F, is placed in a room with a temperature of 78◦ F. If the lemonade warms to 45◦ F in 30 seconds, what will its temperature be in 10 minutes? 11. An iron ingot, heated to a temperature of 300◦ C, is placed in a liquid bath held at a constant temperature of 90◦ C. If the ingot cools to 250◦ C in two minutes, what will its temperature be in 20 minutes? 12. A glass of ginger ale is left in a room. Initially, the ginger ale has a temperature of 45◦ F, but after one minute the temperature has increased to 50◦ F and after two minutes it has increased to 54◦ F.