Question

a function is graphed below. on which interval of x is the average rate of change of the function the smallest?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). We need to calculate this for each interval and compare the values.

Step2: Identify the points on the graph

From the graph, we have the following points (assuming the x - axis intervals and corresponding y - values):
- Let's assume the intervals are \([0,2]\), \([2,6]\), \([6,10]\), \([10,20]\) (from the x - coordinates of the points: \((0,0)\), \((2,3)\), \((6,6)\), \((10,9)\), \((20,15)\))

Step3: Calculate average rate of change for \([0,2]\)

For the interval \([0,2]\), \(a = 0\), \(f(a)=0\), \(b = 2\), \(f(b)=3\).
Using the formula \(\frac{f(b)-f(a)}{b - a}=\frac{3 - 0}{2-0}=\frac{3}{2}=1.5\)

Step4: Calculate average rate of change for \([2,6]\)

For the interval \([2,6]\), \(a = 2\), \(f(a)=3\), \(b = 6\), \(f(b)=6\).
\(\frac{f(b)-f(a)}{b - a}=\frac{6 - 3}{6 - 2}=\frac{3}{4}=0.75\)

Step5: Calculate average rate of change for \([6,10]\)

For the interval \([6,10]\), \(a = 6\), \(f(a)=6\), \(b = 10\), \(f(b)=9\).
\(\frac{f(b)-f(a)}{b - a}=\frac{9 - 6}{10 - 6}=\frac{3}{4}=0.75\) Wait, maybe I misread the points. Wait, looking at the graph again, maybe the points are \((0,0)\), \((2,3)\), \((6,6)\), \((10,9)\), \((20,15)\)? Wait, no, the last point is \((20,15)\)? Wait, the y - axis has 16,15,14,12,10,8,6,4,2,0. Wait, the points are: \((0,0)\), \((2,3)\), \((6,6)\), \((10,9)\), \((20,15)\). Wait, no, maybe the intervals are \([0,2]\), \([2,6]\), \([6,10]\), \([10,20]\)

Wait, let's recalculate:

For \([0,2]\): \(\frac{3 - 0}{2-0}=1.5\)

For \([2,6]\): \(\frac{6 - 3}{6 - 2}=\frac{3}{4}=0.75\)

For \([6,10]\): \(\frac{9 - 6}{10 - 6}=\frac{3}{4}=0.75\)

For \([10,20]\): \(\frac{15 - 9}{20 - 10}=\frac{6}{10}=0.6\)

Wait, but maybe the intervals are different. Wait, the graph has points at (0,0), (2,3), (6,6), (10,9), (20,15). Wait, no, maybe the x - axis is labeled with 0,2,4,6,8,10,12,14,16,18,20,22. And the y - axis with 0,2,4,6,8,10,12,14,16.

Wait, the points are: (0,0), (2,3), (6,6), (10,9), (20,15). Wait, no, the last point is (20,15)? Wait, the y - coordinate at x = 20 is 15? And at x = 10 is 9, x = 6 is 6, x = 2 is 3, x = 0 is 0.

Now, let's calculate the average rate of change for each interval:

1. Interval \([0,2]\):
\(a = 0\), \(f(a)=0\); \(b = 2\), \(f(b)=3\)
Average rate of change \(=\frac{3 - 0}{2-0}=\frac{3}{2}=1.5\)

2. Interval \([2,6]\):
\(a = 2\), \(f(a)=3\); \(b = 6\), \(f(b)=6\)
Average rate of change \(=\frac{6 - 3}{6 - 2}=\frac{3}{4}=0.75\)

3. Interval \([6,10]\):
\(a = 6\), \(f(a)=6\); \(b = 10\), \(f(b)=9\)
Average rate of change \(=\frac{9 - 6}{10 - 6}=\frac{3}{4}=0.75\)

4. Interval \([10,20]\):
\(a = 10\), \(f(a)=9\); \(b = 20\), \(f(b)=15\)
Average rate of change \(=\frac{15 - 9}{20 - 10}=\frac{6}{10}=0.6\)

Wait, but maybe the intervals are shorter? Wait, maybe the points are (0,0), (2,3), (6,6), (10,9), (20,15). Wait, no, perhaps the last interval is \([10,20]\) and the others are \([0,2]\), \([2,6]\), \([6,10]\). Wait, but the question is about which interval has the smallest average rate of change.

Wait, maybe I made a mistake in the points. Let's re - examine the graph:

The first segment: from (0,0) to (2,3) (x from 0 to 2, y from 0 to 3)

Second segment: from (2,3) to (6,6) (x from 2 to 6, y from 3 to 6)

Third segment: from (6,6) to (10,9) (x from 6 to 10, y from 6 to 9)

Fourth segment: from (10,9) to (20,15) (x from 10 to 20, y from 9 to 15)

Now, calculate the slope (average rate of change) for each segment:

- Segment 1 (\([0,2]\)): slope \(=\frac{3 - 0}{2-0}=1.5\)

- Segment 2 (\([2,6]\)): slope \(=\frac{6 - 3}{6 - 2}=\frac{3}{4}=0.75\)

- Segment 3 (\([6,10]\)): slope \(=\frac{9 - 6}{10 - 6}=\frac{3}{4}=0.75\)

- Segment 4 (\([10,20]\)): slope \(=\frac{15 - 9}{20 - 10}=\frac{6}{10}=0.6\)

Since \(0.6<0.75 < 1.5\), the interval with the smallest average rate of change is \([10,20]\) (assuming the last point is (20,15)).

Wait, but maybe the points are different. Wait, the y - coordinate at x = 20 is 15? And at x = 10 is 9, x = 6 is 6, x = 2 is 3, x = 0 is 0.

Alternatively, maybe the intervals are \([0,2]\), \([2,6]\), \([6,10]\), \([10,20]\). So the average rate of change for \([10,20]\) is the smallest.

Answer:

The interval with the smallest average rate of change is \(\boldsymbol{[10, 20]}\) (assuming the function's graph has points \((0,0)\), \((2,3)\), \((6,6)\), \((10,9)\), \((20,15)\)). If the graph has different points, the calculation should be adjusted, but based on the given graph (with the visible points), the interval \([10, 20]\) has the smallest average rate of change.