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Answers without the blur. Sign up and see all textbooks for free! Q. 15

Expert-verified Found in: Page 260 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Explain the difference between two antiderivatives of the function.

The two functions are antiderivatives of each other if their difference is constant.

See the step by step solution

## Step1. Given

The two given functions are $f\left(x\right)\mathrm{and}g\left(x\right)$.

## Step 2. Explanation

$\mathrm{Any}\mathrm{two}\mathrm{functions}\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{derivative}\mathrm{must}\mathrm{differ}\mathrm{by}\mathrm{a}\mathrm{constant}.\phantom{\rule{0ex}{0ex}}\mathrm{Algebraically},\mathrm{this}\mathrm{means}\mathrm{that}\mathrm{to}\mathrm{find}\mathrm{one}\mathrm{antiderivative}\mathrm{of}\mathrm{a}\mathrm{function},\phantom{\rule{0ex}{0ex}}\mathrm{then}\mathrm{all}\mathrm{other}\mathrm{antiderivatives}\mathrm{of}\mathrm{that}\mathrm{function}\mathrm{differ}\mathrm{from}\mathrm{the}\mathrm{one}\mathrm{that}\mathrm{find}\mathrm{by}\mathrm{a}\mathrm{constant}.\phantom{\rule{0ex}{0ex}}\mathrm{Explanation}\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{given}\mathrm{functions}\mathrm{g}\left(\mathrm{x}\right)\mathrm{and}\mathrm{h}\left(\mathrm{x}\right)\mathrm{are}\mathrm{both}\mathrm{antiderivatives}\mathrm{of}\mathrm{some}\mathrm{function}f\left(x\right).\phantom{\rule{0ex}{0ex}}\mathrm{Rewrite}\mathrm{the}\mathrm{functions}\mathrm{as},g\text{'}\left(x\right)=f\left(x\right)andh\text{'}\left(x\right)=f\left(x\right).\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{difference}\mathrm{of}\mathrm{these}\mathrm{functions}\mathrm{is},g\text{'}\left(x\right)-h\text{'}\left(x\right)=0\phantom{\rule{0ex}{0ex}}T\mathrm{his}\mathrm{is}\mathrm{by}\mathrm{the}\mathrm{difference}\mathrm{rule}\mathrm{means}\mathrm{that},\left(8\left(x\right)-h\left(x\right)\right)=0\phantom{\rule{0ex}{0ex}}\mathrm{Use}\mathrm{the}\mathrm{result},"\mathrm{If}\mathrm{is}\mathrm{zero}\mathrm{in}\mathrm{the}\mathrm{interior}\mathrm{of}/\mathrm{then}\mathrm{is}\mathrm{constant}\mathrm{on}/.\phantom{\rule{0ex}{0ex}}\mathrm{Form}\mathrm{the}\mathrm{resul}t,\phantom{\rule{0ex}{0ex}}\mathrm{if}g\text{'}\left(x\right)-\left(x\right)\mathrm{is}\mathrm{zero}\mathrm{then}g\left(x\right)-h\left(x\right)iscons\mathrm{tan}t.\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},g\left(x\right)-h\left(x\right)=C\mathrm{for}\mathrm{some}\mathrm{real}\mathrm{number}C.$ ### Want to see more solutions like these? 