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Solutions for Session 7, Part A
See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6| A7 | A8 | A9 | A10
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Problem A2 | |
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The outputs are: 3, 30, 300, 3,000, 30,000, and 300,000. In each, the input number is the number of zeros in the output. There is a constant ratio between each term. A recursive rule is to multiply by 10 each time to get the next output number.
<< back to Problem A2
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Problem A3 | |
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The graph is increasingly steep as the input value grows, and successive values of the output dwarf all the others. The first output value, 3, is virtually impossible to see if we scale the graph to include the last output value, 300,000.
<< back to Problem A3
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Problem A4 | |
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The input only needs to grow by 4 more to reach 1 billion. The input of 9 gives an output of 3 billion.
<< back to Problem A4
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Problem A5 | |
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The outputs are: 3, 0.3, 0.03, 0.003, 0.0003, and 0.00003. In each, the input number is the number of digits to the right of the decimal point. Again, there is a constant ratio. A recursive rule is to divide by 10 each time to get the next output number.
<< back to Problem A5
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Problem A6 | |
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The graph is similar to the graph in Problem A3, but in reverse: the graph is increasingly flat, and successive outputs are increasingly close to zero. The first output, 3, dwarfs the last output, which is 100,000 times smaller.
<< back to Problem A6
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Problem A7 | |
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The output table can produce neither negative numbers nor zero unless the initial value were negative or zero. This happens because we are multiplying and dividing, step by step, by a positive number, and the only way to yield a negative or zero would be to start with a negative or zero. There are many other positive numbers that do not appear in the tables of Problems A2 and A5, including any number not starting with 3.
<< back to Problem A7
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Problem A8 | |
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The graphs for y = 2x, y = (3/2)x, and y = 8x are increasing. The graph for y = 8x is the steepest graph, while y = 2x is steeper than y = (3/2)x.
The graphs for y = (2/3)x and y = (7/10)x are decreasing. The graph for y = (2/3)x decreases a little faster than y = (7/10)x, but the graphs are much closer together than the increasing graphs. All five graphs pass through the point (0, 1).
<< back to Problem A8
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Problem A9 | |
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The functions y = 2x, y = (3/2)x, and y = 8x were increasing. The functions y = (2/3)x and y = (7/10)x were decreasing. When the base is larger than 1, successive outputs will be larger, because we are multiplying to create a larger number. Additionally, the larger the base (greater than 1), the steeper the graph will be. When the base is between zero and 1, successive outputs will be smaller, since we are multiplying by a number less than 1. Additionally, the smaller the base (closer to zero), the more quickly the graph will advance toward zero.
<< back to Problem A9
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Problem A10 | |
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No exponential function can reverse direction, because the ratio between successive outputs always remains constant. There is an exponential function that never increases or decreases, y = 1x. The function y = 0x is constant when x is positive, but is not defined if x is negative or zero.
<< back to Problem A10
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