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

Question

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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Answer 1

To compare the estimated average rate of change of the exponential function $y=9(13{)}^x$ and the quadratic function $y=9x^2+13x$ 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:

$$\text{Average Rate of Change}=\frac{f(b)-f(a)}{b-a}$$

For the exponential function $y=9(13{)}^x$:

1. Evaluate at $x=0.1$: $$y(0.1)=9(13{)}^{0.1}\approx 9(1.34986)\approx 12.14874$$

2. Evaluate at $x=0.6$: $$y(0.6)=9(13{)}^{0.6}\approx 9(3.422424)\approx 30.801816$$

3. Calculate average rate of change: $$\text{Average Rate of Change}=\frac{30.801816-12.14874}{0.6-0.1}=\frac{18.653076}{0.5}\approx 37.306152$$

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

For the quadratic function $y=9x^2+13x$:

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: $$\text{Average Rate of Change}=\frac{11.04-1.39}{0.6-0.1}=\frac{9.65}{0.5}=19.3$$

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.