Q. 9 - Calculus

Question

Consider the function

$A\left(x\right)={\int }_0^x{t}^{2}dt; B\left(x\right)={\int }_3^x{t}^{2}dt;$

$$\begin{gathered} \left(a\right) \mathrm{The} \mathrm{objective} \mathrm{is} \mathrm{to} \mathrm{explain} \mathrm{why} \mathrm{their} \mathrm{derivatives} \mathrm{differ} \mathrm{by} \mathrm{a} \mathrm{constant} \mathrm{by} \mathrm{comparing} \mathrm{the} \mathrm{derivatives} \mathrm{of} \mathrm{A}\left(\mathrm{x}\right) \& \mathrm{B}\left(\mathrm{x}\right). \\ \left(\mathrm{b}\right)\mathrm{The} \mathrm{objective} \mathrm{is} \mathrm{to} \mathrm{explain} \mathrm{why} \mathrm{their} \mathrm{derivatives} \mathrm{differ} \mathrm{by} \mathrm{a} \mathrm{constant} \mathrm{by} \mathrm{comparing} \mathrm{the} \mathrm{derivatives} \mathrm{of} \mathrm{A}\left(\mathrm{x}\right) - \mathrm{B}\left(\mathrm{x}\right). \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right)\mathrm{The} \mathrm{derivatives} \mathrm{differ} \mathrm{by} \mathrm{constant} \\ \left(\mathrm{b}\right)\mathrm{It} \mathrm{is} \mathrm{signed} \mathrm{area} \mathrm{under} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{y} = {\mathrm{x}}^{(2} \mathrm{from} x = 0 to x = 3 . \end{gathered}$$

Step 1. Given

The given function is $A\left(x\right)={\int }_0^x{t}^{2}dt; B\left(x\right)={\int }_3^x{t}^{2}dt;$

Part(a) Step 2. Explanation

$$\begin{gathered} \mathrm{The} \mathrm{derivative} \mathrm{of} \mathrm{A}\left(\mathrm{x}\right) \mathrm{is} \mathrm{A}\text{'} \left(\mathrm{x}\right)={\mathrm{x}}^2 .. \\ \mathrm{The} \mathrm{derivative} \mathrm{of} \mathrm{B}\left(\mathrm{x}\right) \mathrm{B}\text{'} \left(\mathrm{x}\right)={\mathrm{x}}^2 . \\ \mathrm{Therefore}, \mathrm{the} \mathrm{derivatives} \mathrm{of} \mathrm{A}\left(\mathrm{x}\right) \& \mathrm{B}\left(\mathrm{x}\right) \mathrm{differ} \mathrm{by} \mathrm{a} \mathrm{constant}. \end{gathered}$$

Part(b) Step 3. Explanation

$$\begin{gathered} A\left(x\right)={\int }_0^x{t}^{2}dt; B\left(x\right)={\int }_3^x{t}^{2}dt; \\ \mathrm{Which} \mathrm{means} \mathrm{that} \mathrm{it} \mathrm{is} \mathrm{the} \mathrm{signed} \mathrm{area} \mathrm{under} \mathrm{the} \mathrm{graph} \mathrm{o}f y = x^2 \mathrm{from} x = 0 to x = 3 . \end{gathered}$$