Consider the cubic function d\left(x\right)=\sqrt[3]{2x-1} and the square root function g\left(x\right)=\sqrt{2x+1}. Select the interval where the square root function has the lower estimated average rate of change.
[2, 6]
[3, 7]
[1, 5]
[0, 4]
To find the estimated average rate of change for each function on each interval, we need to find the slopes of their secant lines.
For the square root function g(x) = √(2x + 1):
To find the slope between two points (x1, y1) and (x2, y2), the formula is:
slope = (y2 - y1) / (x2 - x1)
Let's consider the interval [2, 6]:
For x = 2, y = g(2) = √(2(2) + 1) = √(4 + 1) = √5
For x = 6, y = g(6) = √(2(6) + 1) = √(12 + 1) = √13
The slope between these two points is:
slope1 = (√13 - √5) / (6 - 2) = (√13 - √5) / 4
Now let's consider the interval [3, 7]:
For x = 3, y = g(3) = √(2(3) + 1) = √(6 + 1) = √7
For x = 7, y = g(7) = √(2(7) + 1) = √(14 + 1) = √15
The slope between these two points is:
slope2 = (√15 - √7) / (7 - 3) = (√15 - √7) / 4
Comparing slope1 and slope2, we see that (√13 - √5) / 4 > (√15 - √7) / 4.
Therefore, the square root function has the lower estimated average rate of change on the interval [2, 6].