Compare the estimated average rate of change of the exponential function y = 9(1/3)^x and the quadratic function y = 9x^2 and the quadratic function y = 9x^2 + 1/3x. Which of the following function has a negative estimated average rate change over the interval [0.1, 0.6]?
A. the quadratic
B. both functions
C. neither function
D. the exponential function
To find the average rate of change of a function over an interval, we take the difference in the function values at the endpoints of the interval and divide it by the difference in the input values at the endpoints.
For the exponential function y = 9(1/3)^x, the value at the endpoint 0.6 is:
y = 9(1/3)^(0.6) = 9/[(3^0.6)^(0.6)] = 9/(3^0.36) ≈ 9/2.371 = 3.8 (rounded to one decimal place)
The value at the endpoint 0.1 is:
y = 9(1/3)^(0.1) = 9/[(3^0.1)^(0.1)] = 9/(3^0.01) ≈ 9/1.103 = 8.16 (rounded to two decimal places)
The difference in the function values is approximately: 3.8 - 8.16 ≈ -4.36
The difference in the input values is: 0.6 - 0.1 = 0.5
So, the estimated average rate of change of the exponential function over the interval [0.1, 0.6] is: (-4.36)/(0.5) ≈ -8.72
For the quadratic function y = 9x^2, the value at the endpoint 0.6 is:
y = 9(0.6)^2 = 9(0.36) = 3.24
The value at the endpoint 0.1 is:
y = 9(0.1)^2 = 9(0.01) = 0.09
The difference in the function values is: 3.24 - 0.09 = 3.15
The difference in the input values is: 0.6 - 0.1 = 0.5
So, the estimated average rate of change of the quadratic function over the interval [0.1, 0.6] is: (3.15)/(0.5) = 6.3
For the quadratic function y = 9x^2 + (1/3)x, the value at the endpoint 0.6 is:
y = 9(0.6)^2 + (1/3)(0.6) = 9(0.36) + 0.2 = 3.24 + 0.2 = 3.44
The value at the endpoint 0.1 is:
y = 9(0.1)^2 + (1/3)(0.1) = 9(0.01) + 0.03 = 0.09 + 0.03 = 0.12
The difference in the function values is: 3.44 - 0.12 = 3.32
The difference in the input values is: 0.6 - 0.1 = 0.5
So, the estimated average rate of change of the quadratic function over the interval [0.1, 0.6] is: (3.32)/(0.5) = 6.64
Comparing the estimated average rate of change values:
- Exponential function: -8.72
- Quadratic function (y = 9x^2): 6.3
- Quadratic function (y = 9x^2 + (1/3)x): 6.64
Since the estimated average rate of change for the exponential function is negative (-8.72) and the estimated average rate of change for the quadratic functions is positive (6.3 and 6.64), the function with a negative estimated average rate change over the interval [0.1, 0.6] is:
D. the exponential function.