GFT_Task_4_revision

GFT_Task_4_revision - GFT Task 4 Running head: GFT TASK 4...

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GFT Task 4      Running head: GFT TASK 4 CALCULUS I GFT Task 4 Calculus I – Limits and Continuity Western Governors University
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GFT Task 4      Infinite Discontinuity An example of a function with an infinite discontinuity in physics would be something like the one dimensional potential function which looks like . This functions graph is posted below: The horizontal axis is the position of the object relative to the equilibrium point of the spring and mass system. The vertical axis is the value at a particular value. Here we can see that the potential approaches infinity as we move left on the -axis, and heads toward negative infinity as we move right on the -axis. So if we started at the left edge of the picture and went toward the right we would be walking off a cliff, and if we started on the right side and walked left we would be climbing that same cliff on the other side of the axis. That is the essential part of this function being categorized as a function with an infinite discontinuity. The range and domain correspond to the span of the values (the vertical axis), and the span of the values that are generating those values (respectively). Why is it important to say the values generating those values? Because we don't want values that are meaningless to our application (I’ll get to this in a second).
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GFT Task 4      So the range will correspond to the span of the values can be on our graph and in our application. As seen on the graph there are two sets of numbers needed because the function is not continuous ! Formally, a function is continuous at a point x = c if the (standard two-sided) limit exists there and is equal to the value of the function at c . In other words, if f ( x ) = f ( c ) In order to show that f ( x ) = f ( c ) you need to show that f ( x ) exists, and f ( x ) exists, and that they are both equal to f ( c ). A function is considered continuous if it is continuous at all points in its domain. In simple terms, we need a set of numbers (lo,hi) to cover the left side of the graph. These are , which says that the function runs from and that the actual value of and are not included in this set (that's why we use () instead of [] which would be when the value and are actually reached by ). In a similar fashion we see the additional set for the right side of the graph is where again, infinity is not actually reached (which it never is so the "(" or ")" will always be with the sign) nor is actually reached (which does happen in certain functions just not this one). Now this fact of spanning between and happens on intervals called the domain of the -axis, which in our case is representative of the particular positionof an object that we are measuring the of (we will get to this in a second as well). The values that correspond to the
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GFT Task 4      values of the domain mentioned above are for the 's on the left side of the graph (the negative 's) and on the right side of the graph. Notice again the parenthesis instead of brackets on the
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This note was uploaded on 01/30/2012 for the course MATH 1111 taught by Professor Smith during the Spring '11 term at Western Governors.

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GFT_Task_4_revision - GFT Task 4 Running head: GFT TASK 4...

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