AREA AS AN ACCUMULATION PROCESS

Question :

The graph of the function g consists of a quarter-circle and three line segments.

Let f be the function defined by

A. Find f(-5).

B. Find all the values of x on the open interval (5, -4) where f is decreasing. Justify your answer.

C. Write an equation for the line tangent to the graph of f at x = -1.

D. Find the minimum value of f on the closed interval [5, -4]. Justify your answer.

Solution :

Part (A) :

Part (B) :

If f'(x) = 0,

g(x) = 0

In the graph above, g(x) = 0 represents x-intercepts.

The graph intersects x-axis at x = -5, -2, 1 and 4.

When marking these values of x on the number line, we get the following intervals.

(-5, -2), (-2, 1) and (1, 4)

Since we have to find all the values of x on the open interval (5, -4)  where f is decreasing, we have to find the interval at where f'(x) or g(x) is negative.

In the above graph, y or g(x) is negative on the open interval (-2, 1).

That is,

f'(x) < 0 when x ∈ (-2, 1).

Therefore, f is decreasing on the open interval

(-2, 1)

Part (C) :

Find the value of y which is corresponding to x = -1. That is f(-1).