Math 3920 - Appalachian State University



Math 3920

Problem Set 2

1. The table shows the colonial population estimates of the American colonies from 1700 – 1780.

|Year |Population |

|1700 |250900 |

|1710 |331700 |

|1720 |466200 |

|1730 |629400 |

|1740 |905600 |

|1750 |1170800 |

|1760 |1593600 |

|1770 |2148100 |

|1780 |2780400 |

In each of the following, let y represent the population in the year t, with t = 0 corresponding to 1700.

a) Use the regression feature of your graphing calculator to find an exponential model of the data.

b) Use the regression feature of your graphing calculator to find a quadratic model of the data.

c) Use the graphing utility to plot the data and the models from parts a and b in the same viewing window. Sketch what you see and tell what window you chose.

d) Which model is a better fit for the data? Would you use this model to predict the population of the US in the year 2010? 2050? Explain.

2. Linear functions are analogous to which type of sequence? Explain.

3. Exponential functions are analogous to which type of sequence? Explain.

4. Show that the slope-intercept and point-slope equations for lines are equivalent.

5. Fractals

Below are the first three iterations of a fractal. Use these iterations to complete parts a - c below.

a. Above the words “Initial Figure” sketch the figure at n = 0. Draw your initial figure to scale with the other iterations given.

b. Above the words “4th Iteration” sketch the figure at n = 4. Draw the figure to scale with the other iterations given.

c. Describe in complete sentences the recursive process that occurs from one iteration to the next in this fractal.

6. Look at the graph below that depicts Carlos and Maria’s speed as a function of time on their way to school. Answer the following questions with values of time (t). (For example, AB includes all t between A and B.) Use interval notation to write your responses.

[pic]

a. For what values of time (t) are Carlos and Maria stopped?

b. For what values of t do they increase their speed?

c. Which of these intervals show increases at the same rate?

d. For what values of t do Carlos and Maria decrease their speed?

e. At what time (t) do they achieve their highest speed?

f. For what values of t do Carlos and Maria drive at a constant rate?

Calculator Exploration

Clear out all graphs or statistical plots on your calculator. Change the window to [-10, 10] for x and [0, 4000] for y. Graph the following three functions simultaneously:

f(x) = x4-4x3+16x-16

g(x) = x4-4x3-4x2+16x

h(x) = x4+x3-8x2-12x

What can you say about the functions?

Now, change the window to [-5,5] for x and [-35, 15] for y. Graph the same three functions simultaneously. What can you say about the functions?

Discuss their similarities and differences. Which power function do they most closely resemble?

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