\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus Functions - Ceiling and Floor Functions

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Single Variable Calculus
Derivatives
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Multi-Variable Calculus
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Functions

Although the floor and ceiling functions are not used a lot in calculus, you still need to know about them since they will show up occasionally and your calculus instructor will expect you to understand them already. In fact, they will probably not take time to explain them. They will also expect you to understand the notation, so take special note of the notation. Also, they will show up in differential equations, especially the floor or greatest integer function called the step function.

I had a lot of difficulty understanding the ceiling and floor functions. I think part of it was that my instructors didn't explain them very well. They just seemed to wave their hands and expect us to understand them right away. If you can understand them that way, that's great. But not everyone can. It turns out they are not so hard to understand. Here are some videos explaining them in more detail.

Graphs

PatrickJMT - Graphing the Greatest Integer or Floor Function [9min-5secs]

video by PatrickJMT

Floor and Ceiling Functions

Lauren Kuhn - Floor and Ceiling Functions [5min-8secs]

video by Lauren Kuhn

The floor function is used more often than the ceiling function. So here are some additional videos on the floor function.

Bart Snapp - The Greatest Integer Function [4min-8secs]

video by Bart Snapp

MIP4U - The Floor Function (Greatest Integer Function) [8min-18secs]

video by MIP4U

Example

In order to understand the floor function more fully, here is an example where the floor function is part of an equation to solve. Try this on your own before watching the video solution.

Find all \( x \in \mathbb{R} \) such that \( \lfloor x^2+1 \rfloor = \lfloor 2x \rfloor \).

Michael Penn - A quadratic floor equation [13min-26secs]

video by Michael Penn

Really UNDERSTAND Precalculus

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