Lesson Summary - 2.2 PDF

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Colin O’Brien 9/24/17 Lesson Summary: 2.2 - Shh! Please Be Discreet (Discrete)! Concepts We Covered: - Representing arithmetic situations through multiple mathematical representations. An arithmetic situation can be shown as a table, a graph, a recursive equation and an explicit equation. For example, if given the arithmetic sequence {2,4,6,8…} given that n = inputs and f(n) = outputs, I could make a table that would look like this:

Using that table I could plot the points on a graph. Since my table shows that common difference and the starting value I could use both of those to create a recursive and explicit equation. The recursive equation would be f(1) = 2 f(n) = f(n - 1) + 2. The explicit equation would be f(n) = 2 + 2(n - 1). - Representing linear situations through multiple mathematical representations. A linear situation can be shown as a table, a graph, an explicit equation, and not really a recursive equation. For example, if I were given a linear situation like a bucket being filled with water at 2 oz a minute, then n = minute and f(n) = oz in the bucket which I could use to make a table:

Using this table I could plot the points to make a graph that is continuous because the bucket is constantly filling. I could then use the starting value and the slope to make an explicit equation which would be f(n) = 2n. I could make a recursive equation which would be f(0) = 0 f(n) = f(n - 1) + 2 but

since this situation is continuous the recursive equation isn’t the best because it doesn’t represent values in between whole numbers. - Representing geometric situations through multiple mathematical representations. A geometric sequence could be represented through a table, a graph, a recursive equation and an explicit equation. Given the sequence {2,4,8,16…} and n = inputs while f(n) = outputs I could create a table:

Using the table I could make a graph by plotting the points. Then I could use the starting value and the common ratio shown in the table to make a recursive and explicit equation. The recursive equation would be f(1) = 2 f(n) = f(n - 1) x 2. The explicit equation would be f(n) = 2(2)n - 1. - Representing exponential situations through multiple mathematical representations. An exponential situation could be represented through a table, a graph, an explicit equation and sort of a recursive equation. Given the situation in which the bucket filled up to 50 oz and as it was poured out half of it came out each minute. Given that n = minutes and f(n) = oz in the bucket I could make a table looking like this:

Using the table’s points I could plot them and then connect them to make a continuous graph. Using the table’s starting value and the common ratio I could make an explicit equation which would be f(n) = 50(½)n . I could make a recursive equation which would be f(0) = 50 f(n) = f(n -1) x ½ however this isn’t best for a continuous situation because it doesn’t represent values in between whole numbers. - Whether or not a situation is continuous or discrete. Something is discrete if the change is happening at specific intervals. Something is continuous if the

change is happening constantly and consistently. For example, if I were told that the population of the Bay Area grows by 3% each year. I know that the intervals happen consistently at each year but I don’t know about in between each year. Since it happens at each year then it is discrete. If I were given the situation in which the Mississippi River pours water into the Gulf of Mexico at the rate of 100 gallons an hour, I would say it is continuous because the water would flow at a constant rate consistently and always into the gulf. Vocabulary We Learned; - Arithmetic Sequence:* A situation or sequence with a domain limited to natural numbers. For example, the sequence {2,4,6,8…} is arithmetic because it’s domain is {n | natural #s > 0} - Geometric Sequence:*  A situation or sequence with a domain limited to natural numbers. For example, the sequence {2,4,8,16…} is geometric because it’s domain is {n | natural #s > 0} - Step Function Graph: A graph used to show if a situation is discrete or not. For example, if I were given a situation in which a bake sale made $20 every hour, I could make a step function graph to show it is discrete. * We already know these words but were given new definitions....