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Mathematical Expression Editor
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Worked Out Examples Problem Videos
The following videos may be helpful when trying to solve the problems in this practice section. Note that you may skip to the end
of the video to get completion credit for this page if you don’t need to watch them.
Practice Problems: Intermediate Value Theorem (IVT)
How You Can (And Should) Get More Practice!
Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason
you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete
additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight
forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for
you.
Theoretically Easier Difficulty Problem
Remember that IVT needs the function to be continuous, and that there are two values, such that when you evaluate the
function at each, you get a lower number and a higher number than the one you are after.
Since this function is monotonic, try using the ends of the given interval as the two values for IVT.
Let . Use the Intermediate Value Theorem to determine if there exists an -value in the interval such that . If such an -value exists,
enter 1 in the answer box, if not, enter 0. .
Theoretically Medium Difficulty Problem
Remember that IVT needs the function to be continuous, and that there are two values, such that when you evaluate the
function at each, you get a lower number and a higher number than the one you are after.
Since this function is not monotonic, you may need to figure out what some good -values are to try.
Try using, as an value, the value . Can you see why that might be a good choice?
As the other value, try both ends of the interval, and use whichever number is further away from the one you got from the last
hint.
Let . Use the Intermediate Value Theorem to determine if there exists an -value in the interval such that . If such an -value exists,
enter 1 in the answer box, if not, enter 0. .
Theoretically Harder Difficulty Problem
Remember what a horizontal asymptote is - it’s what the function tends to as gets really large.
If you know that there exists at least one -intercept, then you know that must have at least one value where , use this as one of
your values for IVT.
Remember that a function may attain it’s horizontal asymptote - but it may not.
Let be a continuous function with a horizontal asymptote of , and suppose you know that has at least one x-intercept. Must it be
the case that there exists an value such that ? (i.e. do you know that such an value must exist).
If yes, enter 1, if no enter 0:
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)