Q. 14 - Calculus

Question

Let $f$ be the function shown next at the left, and define $A\left(x\right)={\int }_0^{x}f\left(t\right)dt$ On which interval(s) is $A$ positive? Negative? Increasing? Decreasing? Sketch a rough graph of $A$.

Step-by-Step Solution

Answer

If the function $f$ is positive, then the area function $A$ will increasing and if the function $f$ is negative, then the area function is decreasing.

A rough graph of $A$ is,

Step 1 . Given information

$A\left(x\right)={\int }_0^{x}f\left(t\right)dt$.

The given graph is,

Step 2 . From the given figure you observe that, the area of the region is in the positive side of y -axis with the x -axis.

So, the area function is always positive. 

Recollect that, if the function $f$ is positive, then the area function $A$ will increasing and if the function $f$ is negative, then the area function is decreasing.

From the given figure, the function $f$ is positive in the given domain. Thus, $A$ is increasing on the entire domain.

Step 3 . Sketch the rough graph of the area function as shown below: