Compare the estimated average rates of change of the function f(x) = square root of 3x - 4 and g(x) = 2 square root of x - 4/3 over the interval of [2, 3]. State the difference in the estimated average rates of change to the nearest tenth.
To find the average rate of change of a function over an interval, we need to calculate the difference in the y-values (outputs) divided by the difference in the x-values (inputs) over that interval.
For the function f(x) = √(3x) - 4, we need to find the difference in f(x) values between x = 2 and x = 3. Plugging in these values into the function, we get:
f(2) = √(3(2)) - 4 = √6 - 4
f(3) = √(3(3)) - 4 = √(9) - 4 = 3 - 4 = -1
The difference in f(x) values is: -1 - (√6 - 4) = -1 - √6 + 4 = -√6 + 3.
Therefore, the average rate of change of f(x) over the interval [2, 3] is (-√6 + 3) / (3 - 2) = -√6 + 3.
For the function g(x) = 2√x - 4/3, we need to find the difference in g(x) values between x = 2 and x = 3. Plugging in these values into the function, we get:
g(2) = 2√(2) - (4/3) = 2√2 - (4/3)
g(3) = 2√(3) - (4/3)
The difference in g(x) values is: (2√3 - (4/3)) - (2√2 - (4/3)) = 2√3 - 4/3 - 2√2 + 4/3
= 2√3 - 2√2.
Therefore, the average rate of change of g(x) over the interval [2, 3] is (2√3 - 2√2) / (3 - 2) = 2√3 - 2√2.
The difference in the estimated average rates of change of f(x) and g(x) over the interval [2, 3] is:
(-√6 + 3) - (2√3 - 2√2) = -√6 + 3 - 2√3 + 2√2.
We cannot simplify this expression any further, so the difference in the estimated average rates of change of f(x) and g(x) over the interval [2, 3] is approximately -√6 + 3 - 2√3 + 2√2. Rounding to the nearest tenth, the difference is approximately 0.4.