AP Calculus BC · Unit 1: Limits and Continuity
1.13 Removing Discontinuities
In short
This topic focuses on the concept of removable discontinuities. You must understand that a discontinuity is removable if the limit of the function exists at that point. You are expected to know how to define or redefine the function's value to match the limit, thereby making the function continuous.
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What else do you need to know about Removing Discontinuities?
A significant application is with piecewise-defined functions, where you must ensure continuity at domain boundaries by setting the expressions on either side of the boundary equal to each other and to the function's value at that point.
Can you do these?
- I can remove a discontinuity by defining or redefining the value of the function at a point to equal the limit of the function as x approaches that point.
- I can make a piecewise-defined function continuous at a boundary by ensuring the expressions on both sides of the boundary and the function value at the boundary are all equal.
How is Removing Discontinuities tested on the exam?
You are frequently assessed on your ability to find the value of an unknown parameter (e.g., a constant 'k' or 'c') that makes a piecewise function continuous everywhere. This requires setting up an equation where the left-hand limit equals the right-hand limit at the boundary point. The suggested skill is 1.E (Apply appropriate mathematical rules or procedures, with and without technology), indicating that these problems may appear on both calculator and non-calculator sections. A common pitfall is failing to explicitly state that the one-sided limits must be equal to the function value, or making algebraic errors when solving for the parameter.
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Removing Discontinuities: common questions
What do you need to know about Removing Discontinuities for AP Calculus BC?
This topic focuses on the concept of removable discontinuities. You must understand that a discontinuity is removable if the limit of the function exists at that point. You are expected to know how to define or redefine the function's value to match the limit, thereby making the function continuous.
How is Removing Discontinuities tested on the AP Calculus BC exam?
You are frequently assessed on your ability to find the value of an unknown parameter (e.g., a constant 'k' or 'c') that makes a piecewise function continuous everywhere. This requires setting up an equation where the left-hand limit equals the right-hand limit at the boundary point.
How do I know I have mastered Removing Discontinuities?
Check you can do each of these: remove a discontinuity by defining or redefining the value of the function at a point to equal the limit of the function as x approaches that point; make a piecewise-defined function continuous at a boundary by ensuring the expressions on both sides of the boundary and the function value at the boundary are all equal.
Where does Removing Discontinuities fit in AP Calculus BC?
Removing Discontinuities is topic 1.13 in Unit 1: Limits and Continuity of AP Calculus BC.
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