Compare the estimated average rate of change of the exponential function y=9(13)x

and the quadratic function y=9x2+13x
. Which function has a negative estimated average rate of change over the interval [0.1,0.6]
?(1 point)
Responses

neither function
neither function

the exponential function
the exponential function

the quadratic function
the quadratic function

both functions
both functions
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1 answer

To compare the estimated average rate of change of the exponential function y=9(13)x and the quadratic function y=9x2+13x over the interval [0.1,0.6], we need to calculate the average rate of change for each function over that interval.

The average rate of change of a function f over the interval [a,b] is given by the formula:

Average Rate of Change=f(b)f(a)ba

For the exponential function y=9(13)x:

  1. Evaluate at x=0.1: y(0.1)=9(13)0.19(1.34986)12.14874

  2. Evaluate at x=0.6: y(0.6)=9(13)0.69(3.422424)30.801816

  3. Calculate average rate of change: Average Rate of Change=30.80181612.148740.60.1=18.6530760.537.306152

The average rate of change for the exponential function over [0.1,0.6] is approximately 37.31 (positive).

For the quadratic function y=9x2+13x:

  1. Evaluate at x=0.1: y(0.1)=9(0.1)2+13(0.1)=9(0.01)+1.3=0.09+1.3=1.39

  2. Evaluate at x=0.6: y(0.6)=9(0.6)2+13(0.6)=9(0.36)+7.8=3.24+7.8=11.04

  3. Calculate average rate of change: Average Rate of Change=11.041.390.60.1=9.650.5=19.3

The average rate of change for the quadratic function over [0.1,0.6] is 19.3 (also positive).

Summary of Results:

  • Average rate of change of the exponential function over [0.1,0.6]: approximately 37.31 (positive).
  • Average rate of change of the quadratic function over [0.1,0.6]: 19.3 (positive).

Conclusion:

Both functions have a positive estimated average rate of change over the interval [0.1,0.6]. Therefore, the correct response is:

neither function.