Worksheet: Intermediate Value Theorem

In this worksheet, we will practice interpreting the intermediate value theorem and using it to approximate a zero of a function.

Q1:

The function 𝐹 ( 𝑥 ) = 1 𝑥 + 3 satisfies 𝐹 ( 1 ) < 3 and 𝐹 ( 1 ) > 3 . But there is no 𝑥 between 1 and 1 where 𝐹 ( 𝑥 ) = 3 . Why does this not violate the intermediate value theorem?

  • Abecause the function 𝐹 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 𝐹 ( 𝑥 ) = 0 , not 𝐹 ( 𝑥 ) = 3
  • Dbecause the function is not defined on the entire interval [ 1 , 1 ]
  • Ebecause the intermediate value theorem only applies on the interval ( 0 , )

Q2:

The figure shows the graph of the function 𝑓 on the interval [ 0 , 1 6 ] together with the dashed line 𝑦 = 3 0 .

𝑓 ( 0 ) < 3 0 and 𝑓 ( 1 6 ) > 3 0 , but 𝑓 ( 𝑥 ) 3 0 anywhere on [ 0 , 1 6 ] . Why does this not violate the intermediate value theorem?

  • Abecause the intermediate value theorem only applies to cases where 𝑓 ( 𝑥 ) = 0 not where 𝑓 ( 𝑥 ) = 3 0
  • Bbecause the function is not defined on the entire interval [ 0 , 1 6 ]
  • Cbecause the intermediate value theorem only applies to polynomial functions
  • Dbecause the function is not continuous at 𝑥 = 8
  • Ebecause the intermediate value theorem only applies to functions with 𝑓 ( 𝑥 ) < 0 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 .

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