Q. 6 - Calculus

Question

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{function}, A\left(x\right) = {\int }_a^{x}f \left(t\right)dt. \\ \mathrm{Determine} \mathrm{if} \mathrm{A}\left(\mathrm{x}\right) \mathrm{is} \mathrm{increasing} \mathrm{or} \mathrm{decreasing} \mathrm{when} \mathrm{fis} \mathrm{positive} \mathrm{and} \mathrm{decreasing} for x>0. \end{gathered}$$

Step-by-Step Solution

Answer

A(x) will increase. 

Step 1. Given

The given function is 

$A\left(x\right) = {\int }_a^{x}f \left(t\right)dt.$

Step 2. Calculation

The term A(x) is the accumulation of the signed area as x moves from left to right So, if f is positive then the area accumulation A(x) will increase.

Therefore, A(x) will increase.