Q. 15 - Calculus

Question

Explain the difference between two antiderivatives of the function.

Step-by-Step Solution

Answer

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

Step1. Given

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

Step 2. Explanation

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