Given a linear function of the form L(x) = mx + k and an exponential function of the form E(x) = abn for x a real number and constants m, k, a and b, consider the sequence given by l(n) and the sequence given by E(n), where n = 1, 2, 3, 4, .... Both of these sequences can be written recursively:
L(n+1) = L(n) + m and L(0) = k and
E(n+1) = E(n) • b and E(0) = a
The first sequence shows that a linear function grows additively by the same sum m and over equal length intervals (i.e., the intervals between consecutive integers). The second sequence shows that an exponential function grows multiplicatively by the same factor b over equal length intervals (i.e., the intervals between consecutive integers).
An increasing exponential function will eventually exceed any linear function. That is, if f(x ) = abx, is an exponential function with a > 1 and b > 0, and g(x) = mx + k is a linear function, then there is a real number M such that for all x > M, then f(x) > g(x)). Sometimes this is not apparent in a graph displayed on a graphing calculator; that is because the graphing window does not show enough of the graphs for us to see the sharp rise of the exponential function in contrast with the linear function.
Lesson 14 Problem Set Sample Solutions
2. Australia experienced a major pest problem in the early 20th century. The pest? Rabbits. In 1859, rabbits were released by Thomas Austin at Barwon Park. In 1926, there were an estimated billion rabbits in Australia. Needless to say, the Australian government spent a tremendous amount of time and money to get the rabbit problem under control. (To find more on this topic, visit Australia’s Department of Environment and Primary Industries website under Agriculture.)
a. Based only on the information above, write an exponential function that would model Australia’s rabbit population growth.
b. The model you created from the data in the problem is obviously a huge simplification from the actual function of the number of rabbits in any given year from 1859 to 1926. Name at least one complicating factor (about rabbits) that might make the graph of your function look quite different than the graph of the actual function.
3. After graduating from college, Jane has two job offers to consider. Job A is compensated $100,000 at a year but with no hope of ever having an increase in pay. Jane knows a few of her peers are getting that kind of an offer right out of college. Job B is for a social media start-up, which guarantees a mere $10,000 a year. The founder is sure the concept of the company will be the next big thing in social networking and promises a pay increase of 25% at the beginning of each new year.
a. Which job will have a greater annual salary at the beginning of the 5th year? By approximately how much?
b. Which job will have a greater annual salary at the beginning of the 10th year? By approximately how much?
c. Which job will have a greater annual salary at the beginning of the 20th year? By approximately how much?
d. If you were in Jane’s shoes, which job would you take?
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