compare the estimated average rate of change of the exponential function y=\left(9\frac{1}{3}\right)^{x} and the quadratic function y=9x^{2}+\frac{1}{3\ }x. Which function has a negative estimated average rate of change over the interval [0.1, 0.6]?
the exponential function
the quadratic function
neither functions
both functions
To compare the estimated average rate of change of these two functions, we need to find the derivative of each function and evaluate it at the given interval.
For the exponential function y = (9+1/3)^x, the derivative is y' = ln(9+1/3) * (9+1/3)^x = ln(28/3) * (9+1/3)^x.
For the quadratic function y = 9x^2 + 1/3x, the derivative is y' = 18x + 1/3.
Now, let's evaluate the derivatives at the interval [0.1, 0.6]:
For the exponential function:
y' = ln(28/3) * (9+1/3)^0.1 ≈ 0.940899
y' = ln(28/3) * (9+1/3)^0.6 ≈ 4.18783
For the quadratic function:
y' = 18 * 0.1 + 1/3 ≈ 3.3
y' = 18 * 0.6 + 1/3 ≈ 10.9
Since the estimated average rates of change for both functions are positive over the interval [0.1, 0.6], neither function has a negative estimated average rate of change over this interval. The correct answer is "neither functions."