Compare the estimated average rate of change of the exponential function y=9(1/3)x and the quadratic function y=9x2+1/3x. Which function has a negative estimated average rate of change over the interval [0.1,0.6] ?
neither function
both functions
the exponential function
the quadratic function
To find the estimated average rate of change for each function over the interval [0.1,0.6], we need to calculate the change in y divided by the change in x.
For the exponential function, y = 9(1/3)^x, we have:
f(0.1) = 9(1/3)^0.1 ≈ 7.311
f(0.6) = 9(1/3)^0.6 ≈ 2.383
Estimated average rate of change = (2.383 - 7.311) / (0.6 - 0.1) = - 0.973
For the quadratic function, y = 9x^2 + (1/3)x, we have:
f(0.1) = 9(0.1)^2 + (1/3)(0.1) = 0.093
f(0.6) = 9(0.6)^2 + (1/3)(0.6) = 3.54
Estimated average rate of change = (3.54 - 0.093) / (0.6 - 0.1) = 7.894
Since the estimated average rate of change for the exponential function is negative (-0.973) and the estimated average rate of change for the quadratic function is positive (7.894), the exponential function has a negative estimated average rate of change over the interval [0.1,0.6].
Therefore, the correct answer is: the exponential function.