1.     Plot the following graphs together:

 

        What word do the graphs spell out?

 2.     The following questions all refer to the graph of a certain function f, shown in Figure E2 1.

 

a.     Approximately what is f(5)?

 

b.     Approximately what is f(0)?

 

c.     Is the function f invertible? Why or why not?

 

* d. If the function represented the temperature of an alien home world (the horizontal axis representing months, and the vertical axis representing degrees Celsius), about how many months would you say it takes for the alien planet to orbit its sun?

 

Figure E2 1: The graph of a mathematical function f.

 

3.     Create an exponential function that models the population of rabbits on an island. Suppose the population begins at 10, and that after 12 months, there are 1000 rabbits. How many months until the rabbit population exceeds one million? (Hint: Try using f(x) = ae^bx, where a and b are the unknown constants; you'll need the natural log function to solve for b.)

 

! 4.   Suppose the rabbit population in the preceding problem can't grow exponentially because of predators and limited resources. Try modeling the population using a log function, making sure the new function satisfies the original constraints. (Hint: If you run into difficulty solving for the constants, try graphing the log function to see what could be causing the problem.)

 

5.     Say you're building a space-based resource management game and you want the cost of research and development to increase exponentially. Suppose that having spent nothing on research, you want it to cost 10 credits to advance to the next level of development, but that after you have spent 10,000 credits, you want it to cost twice that much to advance to the next level. Develop a function to model this behavior.

 

6.     Prove that you cannot force an exponential function to pass through just any three points. (Hint: Find three points that no exponential function can pass through.)