Q. 3 - Calculus

Question

Let f  be the function that is shown here at the left, and define a new function A so that for every $c\ge 1$, $A\left(c\right)$ is the area of the region between the graph of f and the x-axis over the interval $\left[1,c\right]$. For example, $A\left(2\right)$ is the area of the shaded region in the graph at the right.

From the figures, we can see that f is increasing and positive on $[1,\infty )$ and A is also increasing and positive on $[1,\infty )$. What would you be able to say about the area accumulation function A if f  were instead decreasing and positive? Or increasing and negative? Draw some pictures in your investigation.

Step-by-Step Solution

Answer

While the function of f is increasing, negative on $[1,\infty )$ and the function of A is decreasing, negative on $[1,\infty )$.

While the function of f is decreasing, positive on $(-\infty ,1]$ and the function of A is decreasing, negative on $[1,\infty )$.

Step 1. Given information.

Consider the given question,

For function A, we know for every $c\ge 1$.

Step 2. Draw the functions.

A similar examples can be taken for the functions $$\begin{gathered} f\left(x\right)=-{\left(x-1\right)}^2+1, \\ A\left(x\right)=-2\left(x-1\right) \end{gathered}$$.

Step 3. Interpret the graphs.

Consider the interpretation of the graphs,

The function of f is decreasing over $(-\infty ,1]$.

The function of f is increasing over $[1,\infty )$.

The function of f is positive where $f\text{'}>0$ over $(-\infty ,1]$.

The function of f is negative where $f\text{'}<0$ over $[1,\infty )$.

Also,

The function of f  is decreasing over data-custom-editor="chemistry" $(-\infty ,1]$.

The function of f  is decreasing over $[1,\infty )$.

The function of A is negative where $f\text{'}<0$ over $(-\infty ,1]$.

The function of A is negative where $f\text{'}>0$ over $[1,\infty )$.