Exploration 1.3.1 question 7:

\fExploration 1.4.2; Triangular Differences Involving Sequences of the Form f(n) = ank) Take a look at the sequence of square numbers listed in the first row of numbers below. The first several terms of the sequence are listed and, below that, the first differences of the two numbers immediately above, and below that are the differ- ences of the differences (i.e., the second differences). 16 25 36 49 64 81 ... 11 13 15 17 ... level of 2 nd 2 2 2 2 2 ... difference Notice that for the sequence ne the second differences are all 2. 1. Now explore the differences for the sequence of cubes n'. Do you notice any patterns? notinterested 2. Make a conjecture as to what will happen with the fourth powers and the fifth powers. Make a conjecture about the differences and the resulting constant for the sequence of kth powers of n' where k is a positive integer. [This Exploration can be done by hand, or with a calculator, or in a spreadsheet program such as Excel.] 3. If you are given a sequence of kih powers and, is there a way to figure out the value of the coefficient a? 4. Explain how the triangular differences process is related to the rate of change of your sequence function.4. At Gunterville College, tuition rates for in-state residents were $92 per credit for the first 10 credits, $35 per credit for credits 11-18, and for over 18 credits the rate is $74 per credit. Write a piecewise defined function for the total tuition, T, at Guntersville College as a function of the number of credits, c. Be sure to con- sider reasonable domain and range. + ( c) 92 c if OCS10 920+ 35 (c - 10) if 1146418 1200 + 74 ( c - 8) if (218In Exploration 1.3.1 you explored graphs of distance and time. In question 7' what scenario best describes how the graph was produced? D Morgan started walking quickly to school at a constant rate, then realized she would be too early so she immediately slowed down and continued to school at a constant rate. Q Troy started walking to school at a constant pace, but then realized it was a Saturday, so he turned around and walked back home at the same pace. C} Mark was late for football practice so he ran to the eld at a constant rate. C} Justin started running to school but gradually slowed down on the way because it was a hot Texas day. {3 Amy began walking to school, but then thought she might be late so she started running the rest of the way. In Exploration 1.4.2 you explored the differences involving sequences in the form f (n) = an*. Which of the following is the strongest conjecture for a sequence with the k' power? O The constant difference occurs when a = k. O The constant difference occurs on the k" round and the difference is a . k!. O The constant difference would be 5. O The constant difference occurs on the k" round and the difference is k.In question 4 of Exploration 1.5.3 you explored writing piecewise functions from a given scenario. Based on that function, how much would a student need to pay (in dollars) to take 16 credits at Gunterville College