Question

Let $f$be the function shown earlier at the right, 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 $A$.

Short answer

The graph of $f$is positive from $\left[0,1\right]$and $\left[5,6\right]$so the graph of $A$is increasing from $\left[0,1\right]$and $\left[5,6\right]$.

The graph of $f$is negative from $\left[1,5\right]$so the graph of $A$is decreasing from $\left[1,5\right]$.

The graph of $A$is,

Step-by-step solution

Step 1. Given information

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

The given graph is,

Step 2.The objective is to determine the intervals the graph of A is positive, negative increasing and decreasing.

The graph of $A$is positive on $[0,2]$and negative on $\left[2,6\right]$since the slope of the graph shown is negative on $\left[0,2\right]$and positive on $\left[2,6\right]$.

The graph of $f$is positive from $\left[0,1\right]$and $\left[5,6\right]$so the graph of $A$is increasing from $\left[0,1\right]$and $\left[5,6\right]$.

The graph of $f$is negative from $\left[1,5\right]$so the graph of $A$is decreasing from $\left[1,5\right]$.

Step 3. The graph of A is shown below: