Consider the cubic function d(x)=2x−1−−−−−√3 and the square root function g(x)=2x+1−−−−−√ . Select the interval where the square root function has the lower estimated average rate of change.(1 point) Responses
A. [0,4]
B. [1,5]
C. [3,7]
D. [2,6]
To compare the estimated average rate of change of the two functions, we can look at the slopes of the lines tangent to the functions at various points within the given interval.
For the square root function g(x) = 2√x + 1, the estimated average rate of change can be approximated by the slope between two points on the curve. Let's consider two points, (x1, g(x1)) and (x2, g(x2)), within the interval.
The slope between these two points is given by m = (g(x2) - g(x1))/(x2 - x1).
For the cubic function d(x) = 2x - √3, the estimated average rate of change can be approximated by the slope of the line tangent to the curve at a given point. This slope can be obtained by taking the derivative of the function and evaluating it at a specific point.
The derivative of d(x) = 2x - √3 is d'(x) = 2.
Since the derivative is constant for the cubic function, the estimated average rate of change is constant throughout the interval for the function d(x). The value of the estimated average rate of change is 2.
In comparison, for the square root function g(x), the estimated average rate of change varies depending on the choice of two points within the interval.
To determine the interval where the square root function has the lower estimated average rate of change, we need to consider two points in the interval and calculate the slopes between them.
Let's consider the interval [0,4].
If we choose (0, g(0)) and (4, g(4)), the slope is:
m = (g(4) - g(0))/(4 - 0) = (2(4)^0.5 + 1 - (2(0)^0.5 + 1))/(4 - 0) = 5/4 = 1.25.
Therefore, the slope for the square root function within the interval [0,4] is 1.25, which is lower than the constant estimated average rate of change of 2 for the cubic function.
Hence, the square root function has the lower estimated average rate of change in the interval [0,4].
Therefore, the answer is A. [0,4].