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Unit 1

1.16 Working with the Intermediate Value Theorem (IVT Calc)

2 min readโ€ขfebruary 2, 2021

Anusha Tekumulla


The Intermediate Value Theorem (IVT)

The intermediate value theorem focuses on a crucial part of continuity: for any function f(x) that's continuous over the interval [a, b], the function will take any y-value between f(a) and f(b) over the interval.ย ๐Ÿ“ˆ

Specifically, it means that for any value L between f(a) and f(b), there's a value c in [a,b] for which f(c) = L.ย 

While this may sound confusing, it is actually very simple. As we have discussed before, continuous functions have to be drawn without lifting the pencil. If we know the graph passes through (a, f(a)) and (b, f(b)) then it must pass through any y-value between f(a) and f(b).

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fdownload.svg?alt=media&token=0c354704-9bca-401f-8cc4-204f59de4365

Image courtesy of Math is Fun

As you can see in the graph above, you can use the intermediate value theorem to prove that a function will cross a line at least once in an interval [a, b].ย ๐Ÿ‘

Note: The IVT fails if f (x) is not continuous on [a, b].

The IVT is a powerful tool, but it has its limitations. Take a look at this example from OSU about what the IVT can and cannot tell us.

IVT FRQ Practice

In 2012, the Intermediate Value Theorem was the topic of an FRQ. Fin the full text of the prompt here courtesy of mathisfun.com. In summary, the Intermediate Value Theorem says that if a continuous function takes on two values y1ย and y2ย at points a and b, it also takes on every value between y1ย and y2ย at some point between a and b.ย 

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