Today in Calculus

1
Today in Calculus

• Notes
• Continuous Functions
• Identifying types of discontinuity
• Removing removable discontinuities
• Intermediate Value Theorem
• Go over quiz
• Homework

2
Continuous Functions Definition

• Interior point A function is continuous at an
  interior point c of its domain if
• Endpoint A function is continuous at a left
  endpoint a or right endpoint b of its domain if
• If a function is undefined at a point, then its
  discontinuous at that point.

a
b
3
Removable Discontinuity

4
Oscillating Discontinuity

• It oscillates too much to have a limit as x?0

5
Jump Discontinuity
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Infinite Discontinuity
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Discontinuities
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Finding points of discontinuity

• Where does the function not exist?
• Find VA and holes
• Use graph or table

9
Removing a discontinuity

1. Factor and find holes
2. Find limit at each removable point, set

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Removing a discontinuity

• Is the function continuous at x1?
• What value should be assigned to f(1) to make the
  extended function continuous at x 1?

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Intermediate Value Theorem

• A function y f(x) that is continuous on a
  closed interval a, b takes on every value
  between f(a) and f(b). In other words, if y0 is
  between f(a) and f(b), the y0 f(c) for some c
  in a, b.

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Homework

• Pg 80 1,7,9,11-16,21,24,25,26,35-38
• Bring calculators tomorrow