Q. 15 - Calculus

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 $A$.

Step-by-Step Solution

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 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: