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Worksheet: The Intermediate Value Theorem
Q1:
The function satisfies and . But there is no between and 1 where . Why does this not violate the intermediate value theorem?
- Abecause the function is not continuous over its domain
- Bbecause the intermediate value theorem only applies to polynomial functions
- Cbecause the intermediate value theorem only applies to cases where , not
- Dbecause the function is not defined on the entire interval
- Ebecause the intermediate value theorem only applies on the interval
Q2:
The figure shows the graph of the function on the interval together with the dashed line .
and , but anywhere on . Why does this not violate the intermediate value theorem?
- Abecause the intermediate value theorem only applies to cases where not where
- Bbecause the function is not defined on the entire interval
- Cbecause the intermediate value theorem only applies to polynomial functions
- Dbecause the function is not continuous at
- Ebecause the intermediate value theorem only applies to functions with at some value
Q3:
The function is defined on the interval and is continuous there. It is known that and , and these are the only values of with . It is also known that . Explain why .
- Abecause and we already know the two values where it is equal to 3
- Bbecause is an increasing function
- Cbecause should be greater than or equal to
- Dbecause if , then would equal 3 at some point between and by the intermediate value theorem
Q4:
The figure shows only a part of the graph of the function , which is defined on all of .
If we say that for every , what can we conclude about ? Why?
- Athe function is differentiable, because the parts shown look like that
- Bthe function is continuous, because the parts shown look like that
- C for and for , because of the way the graph is drawn
- Dthe function is not continuous, because of what the intermediate value theorem states
- ENo conclusion is possible with the information given.
Q5:
The figure shows only parts of the curve .
We know that the function has the following properties: , is continuous, , and . By considering the difference , what can you conclude about this function?
- AThe function takes the value 0.4 at some point.
- BThere is no conclusion possible.
- CThe function has an inflection point somewhere.
- DThere must exist a point such that .
- EThe function is zero at some .