On the interval given:
the function above is decreasing, which means that the function's average rate of change is negative
the function above is concave-down, which means that the function's average rate of change is decreasing
We discussed rates of change, increasing/decreasing functions, and concavity
in Topic 3,
and we are revisiting these concepts now. The thing to remember is that there
are two DIFFERENT questions to ask:
1. Is the function increasing or decreasing?
2. Is the function concave-up or concave-down?
These two questions are answered by different aspects of the function's rate-of-change.
Example 1:
On the interval given:
the function above is decreasing, which means
that the function's average rate of change is negative
the function above is concave-down, which means
that the function's average rate of change is decreasing
Example 2:
On the interval given:
the function above is increasing, which means
that the function's average rate of change is positive
the function above is concave-up, which means
that the function's average rate of change is increasing
Example 3:
On the interval given:
the function above is increasing, which means
that the function's average rate of change is positive
the function above is concave-down, which means
that the function's average rate of change is decreasing
Example 4:
On the interval given:
the function above is decreasing, which means
that the function's average rate of change is negative
the function above is concave-up, which means
that the function's average rate of change is increasing
A piecewise linear function is one whose graph consists of
pieces of different lines:
Sometimes, a piecewise linear function is a step function:
A function that is represented symbolically as y = c / x , where c is a constant, y represents the dependent variable, and x represents the independent variable, is an inversely proportional function. As x increases, y decreases, and vice-versa. Inversely proportional functions are NOT linear.
In the above inversely proportional function, consider the rate of change at several pairs of points. Are all these rates of change negative? As you move to larger x values, do the rates of change get less negative (increase?). Can you make any conclusion about the concavity of the function if the rate of change of the function is increasing?